multifrequency holography using backpropagation

TR&SOI~IC IMAGING 8, X13-224 (1986)

MULTIFREQUENCY HOLOGRAPHY USING BACKPROPAGATION Tat-Jin Teo and John M. Reid

Biomedical Engineering and Science Institute Drexel University

Philadelphia, PA 19104

The technique of wavefield backpropagation has been used quite extensively in the literature. We report on an analytical study of the resolution properties of this technique. Backpropagation as a form of holographic reconstruction suffers from poor axial resolution. We derive expressions for both the axial and the lateral resolutions. We also show that the axial resolution can be substantially improved by the use of multiple frequencies. We derive an expression relating the resolution and bandwidth. 0 1986 Academic Press, Inc.

Key words: Angular spectrum decomposition; backpropagation; broadband tech- niques.

1. INTRODUCTION Backpropagation [I-G] is holographic in nature since both amplitude and

phase, which are available from measurement, are used. It has many characteristics srmilar to that of the more traditional form of holographic reconstruction. In this paper, we study one of these characteristics, namely, the relatively poor axial resolution compared to the lateral resolution. We will see that the axial resolution is very much aperture-limited and that by the use of multiple frequencies, we can improve the axial resolution substantially. In section 2, the theory of backpropagation is reviewed. In section 3, we discuss the single frequency expressions for both the axial resolution and the lateral resolution. We see that for infinite apertures, the resolution is similar in both the axial and the lateral directions. How- ever, for a finite aperture, the axial resolution is substantially worse than the lateral resolution. In section 4, we show that the axial resolution can be substantially improved with multiple frequencies and we derive a new expression relating the resolution, measured as the size of the axial field of a reconstructed point source, and bandwidth. A discussion of the simulation results and our conclusions are presented in section 5.

2. THEORY OF BACKPROPAGATION The method of angular spectrum decomposition

a field from one plane to another. A given field is cl 71 is used to backpropagate ecomposed into its angular

spectrum, which is a set of plane waves traveling in different directions. The propagation of each plane wave component is then modeled as a linear shift- invariant filtering operation which has a particularly simple transfer function in the form of a multiplicative exponential term. After each plane,wave component has been either forward propagated or backward propagated to another plane, the field on the plane can be synthesized from the set of modified plane waves. This procedure is intrinsically monochromatic, but can be extended to arbitrary waveform by an additional Fourier transform.

The above procedure may be summarized as follows: 1. Fourier transform the field t,o obtain the angular spectrum. 2. Propagate each component forward or backward by multiplying by an appro-

priate propagation filter. 3. Inverse Fourier transform to obtain the field on the second surface.

The multiplicative exponential filter term has two different forms depending on whether the particular components of the angular spectrum that we are propagating have a spatial frequency greater or smaller than the wavenumber, k. Those components that have spatial frequencies greater than the wavenumber are

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TEO AND REID

collectively known as evanescent waves. The other components are called propagating waves or nonevanescent waves. In the literature, they are also known as inhomogeneous and homogeneous waves, respectively.

In appendix A, we present the propagation filter and, with different inversion schemes, we obtain three slightly different forms of the backpropagation filter. The first form is an exact inverse filter which has a positive exponential in the reconstruction and accounts for the evanescent waves. The second form is the complex conjugate of the forward filter rather than the inverse and as such has a negative exponential for the evanescent waves in the reconstruction. Finally, the third form neglects all evanescent waves and only inverts the propagating waves.

When the exact inverse filter is used, the reconstruction of a point source is ideal in the sense that the reconstructed field has a singularity at the source location when an infinite aperture is used. In real life situations, where data are collected at large distances compared to the wavelength, the evanescent waves would most likely be buried in noise and the exact reconstruction is therefore unreliable. The second form avoids the problem of amplifying the noise but it is not clear whether any information carried by the evanescent waves can be restored. The third form has been chosen as the preferred form to use in the reconstruction algorithm. The effect of not using the evanescent waves limits the resolution to the order of the wavelength.[8]

3. RESOLUTION In appendix C, we derive the expression for the reconstruction of a point

source using both the exact inverse filter and the complex conjugate filter. This derivation makes use of the angular spectrum expansion of the free space Green’s function of the Helmholtz equation. This expansion is generally attributed to H. Weyl [15]. We give a derivation of the expansion in appendix B, following the outline given in [13], for the purpose of completeness.

We see that while the exact inverse filter reproduces the point source with its singularity at the origin, the integral in Eq. (C-5) diverges when we backpropagate past the source point. On the other hand, we cannot obtain a closed form expression for the complex conjugate filter. The homogeneous filter, which ignores the.evanescent waves, reconstructs a diffraction-limited image of the point source without the singularity at the origin but with a magnitude proportional to the wavenumber. In appendix D, the same derivation is done for a finite circular aperture. We see that the axial resolution and the lateral resolution are affected very differently by the aperture size.

In the first approximation, the lateral resolution depends linearly on the radius of the aperture while the axial resolution depends on the solid angle subtended by the aperture at the source point. The well known expression for the lateral resolution is

r - abe, - a

if the first zero crossing is used to measure the resolution or

if the 3-dB point is used to measure the resolution. Here a is the radius of the circular aperture and d is the distance between the point source and the collect- ing plane. We have also derived an expression for the monochromatic axial resolution

z = 2.76; (lb)

Here X is the wavelength of the ultrasonic waves and 0 is the solid angle subtended by the aperture at the source point.

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MULTIFREQUENCY HOLOGRAPHY USING BACKPROPAGATION

Using the first zero crossing as a rough measure of resolution, we see that for a circular aperture with a radius of 8 wavelengths and placed at a distance of 65 wavelengths, we have a lateral resolution of approximately 2 wavelengths but an axial resolution of about 60 wavelengths. If we double the radius of the aperture, the lateral resolution will be twice as good but the axial resolution will improve almost three times. The above two expressions should be compared with Eqs. (C- 9) and (C-10) which are for infinite apertures. 4. RESOLUTION AND BANDWIDTH

The axial resolution can be improved by a number of methods. In diffraction tomography, the axial resolution is poor compared to the lateral resolution, and by using many views, the overall resolution can be much improved.(5,9] However, we know that in a typical B-scan image, it is the lateral resolution which is worse than the axial resolution. The latter is determined by the pulse width of the insonification. Hence, if we perform the backpropagation with a wide band insonification, we should be able to obtain much better axial resolution. The use of multiple frequencies in conventional acoustical holography has been reported in the literature [lO,ll] and we have done some preliminary studies in applying multiple frequencies to backpropagation. These studies indicate that multiple frequencies improve axial resolution.[9,12]

In appendix D, we also derive the expression relating the axial field of a reconstructed point source and bandwidth when we use a wideband signal with a uniform spectrum. This expression will allow us to predict the bandwidth needed to do the backpropagation in order to obtain a particular axial resolution. How- ever, the expression is a complicated one, and it is difficult to see the dependence of the resolution on the bandwidth. In figure 1, we have plotted the expression using bandwidth as a parameter. The bandwidth is expressed as percentage bandwidth of the center frequency which is 1 MHz. The X axis is the axial distance in cm and the Y axis is the normalized pressure magnitude. The unlabeled horizontal line indicates the 3-dB resolution width. 5. DISCUSSION AND CONCLUSION

We have performed a computer simulation to test our prediction of the relationship between the axial resolution and bandwidth. The theoretical predictions are displayed in figure 1 while those obtained with computer simulation are

-3 -2 -1 0 1 2 3

Reconstructed axial field magnitude of a point source at the origin with data collected on a circular aperture perpendicular to the axis. Percen- tage bandwidth is used as a parameter. The center frequency is 1 MHz. (theory)

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TEO AND REID

N R 0 e

2

‘d /

0.6 F t

I

0.2 1

0.0 L cm1

-3 -2 -1 0 1 2 3

Reconstructed axial field magnitude of a point source at the origin with data collected on a circular aperture perpendicular to the axis. Percen- tage bandwidth is used as a parameter. The center frequency is 1 MHz. (simulation)

displayed in figure 2. The simulations performed here were done using a two- dimensional planar array while those reported in our preliminary studies[9,12] were done with a one-dimensional linear array. Since a set of discrete frequencies is used rather than a continuous distribution of frequencies, there will be multiple images of the sources, or aliases, in the reconstructed image with the distance between aliases determined by the spacing interval of the discrete frequencies assuming that uniform frequency spacing is used. These aliases can be explained intuitively by considering the synthesis of a wide band pulse with a finite set of discrete frequencies. The result of the synthesis would be a periodic pulse train. The separation between the pulses, or aliases as they are called here, can be increased so that they lie outside the region of interest by reducing the interval

X---Slmulatlon

w 1.5

A

k 1.0

L c m

0.5

0.0 --- I Percent !3 ar~-iu~dth

0 10 20 30 90

Fig. 3 Comparison of the 3-dB axial resolution.

216


among the discrete frequencies. The computer simulation showing how the reconstructed image is affected by bandwidth and the number of discrete frequencies spanning the bandwidth has been reported in our preliminary study.[I2\

Using the width between 3-dB points as the measure of resolution, we com- pare the theory with the simulation in figure 3. We have obtained fairly good agreement between the theory and the computer simulation and we have shown that the use of multiple frequencies can improve the axial resolution of the backpropagation technique substantially.

ACKNOWLEDGEMENTS We have the pleasure to acknowledge the many helpful discussions

with Professor Oleh Tretiak of the Electrical and Computer Engineering Depart- ment at Drexel University and the assistance of Dr. Anthony Devaney in editing the mathematics. This work was supported by funds from the Ernest N. Calhoun chair and a research grant from the National Institutes of Health, # HL 30045.

References

Boyer, A. L., Hirsch, P. M., Jordan, J. A. Jr., Lesem, L. B. and Van Rooy, D. L., Reconstruction of Ultrasonic Images by Backward Propagation, in Acoustical Holography 3 , A. F. Metherell, ed., 333-348 (Plenum Press, New York, 1971).

Powers, J. P., Computer Simulation of Linear Acoustic Diffraction, in Acoustical Holography 7 , L. W. Kessler, ed., 193-205 (Plenum Press, New York, 1978).

Higgins, F. P., Norton, S. J. and Linzer, M., Optical Interferometric Visuali- zation and Computerized Reconstruction of Ultrasonic Fields J. Acoust. Sot. Amer. 68, 1169-1176 (1980).

Stepanishen, P. R. and Benjamin, K. C., Forward and Backward Projection of Acoustic Fields Using FFT Methods, J. Acoust. Sot. Amer. 71 , 803-812 (1982).

Devaney, A. J., A Filtered Backpropagation Algorithm for Diffraction Tomography, Ultrasonic Imaging 4 , 336350 (1982).

Williams, E. G., Maynard, J. D. and Skudrzyk, E., Sound Source Recon- structions Using a Microphone Array, J. Acoust. Sot. Amer. 68 , 340-344 (1080).

Goodman, J. W., Introduction to Fourier Optics, ch. 9 (McGraw-Hill, New York, 19G8).

Williams, E. G. and Maynard, J. D., Holographic Imaging Without the Wavelength Resolution Limit, Phys. Rev. Lett. 45, 554-557 (1980).

Teo, T. J. and Reid, J. M., Spatial / Frequency Diversity in Inverse Scatter- ing, in Proceedings of IEEE 1985 Ultrasonics Symposium 2 , pp. 800-803, IEEE, 1985

Ermert, H. and Karg, R., Multifrequency Acoustical Holography, IEEE Tran. Sonics and Ultrason. B6 , 279-28G (1979).

Nagai, K., Multifrequency Acoustical Holography Using a Narrow Pulse, IEEE Tran. Sonics and Ultrason. 31 , 151-156 (1984).

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Teo, T. J., and Reid, J. M., Angular Spectrum Decomposition : Improving the Resolution, in Acoustical Imaging 14 , A. J. Berkhout, J. Ridder and L. FL van der Wal, eds., 143-154 (Plenum Press, New York, 1985).

Banos, A. Jr., Dipole Radiation in the Presence of a Conducting Half-Space, C/L 2 (Pergamon Press, Oxford, 196G).

Berkhout, A. J., Seismic Migration vol. A, appendix F, third edition (Elsevier, Amsterdam, 1985).

Shewell, J. R. and Wolf, E., Inverse Diffraction and a New Reciprocity Theorem, J. Opt. Sot. Amer. 58, 1596lGO3 (19GS).

Wolf, E. and Shewell, J. R., The Inverse Wave Propagator, Phys. Lett. 25A, 417-418 (1967).

Gradshteyn, I. S. and Ryzhik, I. M. Table of Integrals, Series, and Products, 882 (Academic Press, New York, 1980).

Appendix A Backpropagation Filter

The transfer function for the forward propagation filter is given by

G (u ,v ;d )= e id %/KG%7

e -d v’uw u”+v’_<k2 u2+v2>k2

(A-1)

where d is the distance of propagation and k is the wavenumber. u ,v are the spatial frequency domain variables that correspond to x and y directions, orthog- onal to d, respectively.

1) The inverse filter

G-‘(u ,v ;d ) = e -id dm2

e +d m u2+u2<k2 u2+v2>k2

2) The conjugate filter

G * (u ,v ;d ) = e -id m

,-dd- u2+v2<k2 u2fv2>k2

3) The homogeneous filter

Gh (u ,v;d) = e -id v’m u2+v2sk2

0 U2fV2>k2

(A-2)

(A-3)

(-4-4)

Appendix B 2-D Spatial Fourier Transform of a Spherical Wave Field

Consider the following inhomogeneous Helmholtz equation 02g +k2g = -47rS(r-r,,), P-1)

where 6(r-re) is the Dirac delta function representing the point source at location ro. The free space Green’s function is g. Taking 3-D Fourier transform on both

218


sides of the equation, we have,

-(u2+w2+w2)G+k2G = -4pi, (B-2) where G is the Fourier transform of g and u, v, w are the spatial frequency domain variables corresponding to x, y and z respectively.

The inverse Fourier transform of G in spherical coordinates can be written as

g (x ,Y 1% )=G,,G (R-3)

where h2=u 2+v 2+-w 2 . Evaluating the above triple integral, we obtain the Green’s function, g,

which is the spherical wave field from a point source, namely, exp(ikR)/R. To obtain the 2-D spatial Fourier transform of the spherical wave field in a plane perpendicular to z , we express the above inverse Fourier transform in Cartesian coordinates and evaluate the integration with respect to w using contour integration :

(R-4)

where r2=k 2-u 2-v 2 . The contours are indicated in figure 4 with the two poles at w =7 and

w ‘“y.

Other integral representations of the Green’s function can be found in 1131 and an alternative derivation of the above result is given in [14]. The above result appeared in [15-161 and is generally attributed to H. Weyl [15].

Fig. 4 Contour integration with respect to w.

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TEO AND REID

Appendix C Resolution with Infinite Apertures

To derive the resolution expression for an infinite aperture, we consider a point source located at the origin in figure 5. The field at plane 2 is given by

u 2(x IY 72) =

Using the result in appendix B, we have,

(C-1)

(C-2)

where p2=, 2fv 2 . 1. Using the inverse filter G-’ :

We backpropagate a distance of z r and find

where u =pcost?,v =psine,x =Rcos q!~,y =Rsin q5 .

For lateral resolution, we have 1 z2 1 = ( z r ( . Using the integral expression for the zero order Bessel function of the first kind, we can reduce the double integral in Eq. (C-3) to a single integral as in Eq. (C-4) :

(C-4)

/

Plane 1

/

t

Reconstruction

Pl.Sle

Plane 2

Fig. 5 Geometry used with the derivation for infinite aperture.

220


where t is defined as p . k

We can further simplify Eq. (C-4) to

i sinkR coskR e ikR

u I(5 ,y ;O)=_+-=-, R

where the evaluations of the definite integral involving Bessel functions were found in [17].

For axial resolution, x=0, y=O; i.e. R = 0.

co e -&kz u l(o,o;z )=iki ‘g tdt +k/ m

1

tdt =ikse iskz ds +ky e -” kz ds’ = o 0

where z==1z21-IzLI;s=~ands’ =a . Since the last integral diverges for 2 <0 , we only consider absolute values of z . It is thus difficult to speak about axial resolution using the exact inverse filter.

2. Using the conjugate filter G ’ : We backpropagate a distance of z , and find

u 1(z ,y ;z )=ikj e i d’i?kz ~e-m~l~21+l~~/~

O&P J,( tkR )tdt +kJ

dz J,(tkR )td&X)

1

A closed form expression can not be found for the second integral. However, if and R=O, we can evaluate the magnitude of the reconstructed

,

I 2~ IWW) I =“$e+kI e -2mk ( z2 1

a tdt = I ik++ I (C-7) 2

3. Using the homogeneous filter G * : We backpropagate a distance of z r and find

1

u 1(z ,y ;z )=ikJ ,idi?kz

o TJ0ttn-R W

For lateral resolution : z=O.

i sinkR u 1(x ,Y ;0)=7

Using first zero crossing to measure the resolution, we have

R = 0.5 X,

or using the 3-dB point to measure the resolution, we have

R =0.22X.

For axial resolution, x=0, y=O; i.e. R =0 :

ikz sink u l(O,O;z )=d e z tdt =e TG---$-

Using the 3-dB point to measure the resolution, we have

t = 0.44 x

(C-8)

(C-9)

(C-10)

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TEO AND REID

Appendix D Resolution with Finite Aperture

From appendix B, we have the Fourier transform of the free space Green’s function. It is clear that the inverse Fourier transform of the homogeneous backpropagation filter is given by

e -id dm = F -1 2d[

1 as* _ ---I 2n a.2

(D-1)

Using the convolution property of Fourier transform, the product of the angular spectrum and the transfer function of the propagation filter can be written as the convolution of the field and dg * /as .

d rprime , (D-2)

where r=(z ,y ;O) and r’ =(z ,y ;z ) as indicated in figure 6. The partial deriva- tive is to be evaluated at z =d .

From Eq. (D-2), we have

21 l(r) = -$y?u 2(r’ )(-ik - 00

To simplify the above expression, we make the following assumptions:

1 r-r’ 1 z 1 r’ 1 =(R 2+d 2)1/2 in the denominator, and

where R2 is (CC’ )2+(y’ )2 and ?I, is the angle between r andR .

Then

RdRd $. (D-3)

Fig. 6 Geometry used with the deriva tion for finite circular aperture.

222


If we consider a finite aperture of radius a , we have,

RdR , (D-4)

where J,,(z ) denotes the Bessel function of the first kind and order zero and whose integral expression is used in obtaining eq.(D-4). If a <<d , we can evaluate the integral approximately and obtain

The first zero of Jr(z ) is 3.8318, and we obtain an expression for the lateral resolution as follows:

r = 0.61= a (D-5)

when using the first zero crossing to measure the resolution or

r = 0.27s a,

when the -3-dB point is used to measure the resolution. From eq.(D-4), we can evaluate the magnitude of the reconstructed point

source by setting r to zero. We obtain the following expression :

u ,(o,o;o)+, P-6)

where R is the solid angle subtended by the aperture at the source point and X is the wavelength of the insonification.

To obtain the axial resolution, we use Eq. (C-10) from appendix C but replace the upper limit with sine and obtain

‘me e i JT-l?kkz u &l,O;z)= ; m tdt = +[e rkz-, tkzcose]

where 8 is as indicated in figure 6.

7) After some algebraic manipulation, we obtain for the magnitude of Eq. (D-

nsinu 1 u 1(O,O;z ) 1 = f =,;; n, = xzt’

.zfl where u = - . 2x

It should be noted that when we set .Z equal to zero, we again obtain the magnitude of the reconstructed point source. This result is obtained from expressions derived for the spatial frequency domain and it agrees with that of Eq. (D- 6) which was derived for the space domain.

The expression for the 3-dB axial resolution is

2.761

z=-ii-’

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TEO AND REID

To improve the axial resolution,, we propose the use of multifrequency backpropagation. We integrate expression D-7 with respect to frequency over a bandwidth of A f centered at f m .

s1n* . rrzaj cod e

i Przf m C e

i 3~2.f m c0se sin c

= z KZA~ - P TZ A f cosB

c C

The above expression provides the relationship between the reconstructed axial field of a point source and bandwidth. By varying the bandwidth as a parameter, one can see that the resolution capability of the backpropagation technique can be greatly improved. The plot of the above expression varying bandwidth as a parameter is given in figure 1 in the text.

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multifrequency holography using backpropagation

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