Hologram aperture synthesis with frequency sweeping

Hologram aperture synthesiswith frequency sweeping G. BERBEKAR and S. T£)KI~S

A simple principle is presented to decrease the number of detectors used for mapping acoustical holograms. The diffraction pattern of an object illuminated with coherent radiation compresses with increasing frequency. This is equivalent to the case when the detectors move along the diffraction pattern. This phenomenon is not restricted to the field of acoustics, but can also be utilized in optics and microwaves.

Introduction Here we present a new method of achieving faster hologram aperture scanning by changing the frequency of the sound. This is done without mechanical motion to minimize the number of the elements in the detector matrix. In the following section the physical principle of this method is shown in a very simple case. The next section concerns the case of Fourier and Fraunborer holography. Following that we discuss some difficulties arising from the need for coherent illumination for holographic processes. In the section after, some examples illustrate how and where this method can be used. In the last section we show a few examples of our computer modelling. The detection of an acoustic wavefront with microphones is self-evident, but rather expensive, and therefore several other methods have been developed to convert an acoustic wavefront into optical or electrical signals for further processing, such as liquid surface deformation, chloresteric liquid crystals, solid surface detectors, electron or laser scanning, knife-edge methods etc. 1'2

the scattered acoustic wavefront moves in front of the array of receivers. 2 It is shown in this paper that mechanical scanning can be replaced by sweeping the frequency. The hologram or the Fourier spectrum of an ordinary object contains information about unimportant details. 3-s This fact is utilized in picture processing and faltering. So in many cases of acoustical holography, it is satisfactory only to pick up predetermined parts of the hologram. This has the advantage that the number of receivers (or processing time) can be decreased.

A simple physical example Let us regard a slit, infinite in one dimension, illuminated with a coherent plane wave (Fig. 1). The intensity distribution at the plane S can be expressed as 7

,(x) : K_ [sin(2,xd/zX)] 2 X [ 2rrxd/zX J

5 >

The main purpose of these investigations mentioned above is to eliminate the expensive ultrasonic receivers. A group of methods, known as synthetic apertures, are widely used in acoustic testing and imaging systems. The advantage of this method is that a greater area can be mapped with fewer receivers, at the expense of longer processing time. In one of these techniques one (in longwave acoustical holography) or several receivers arranged in a linear array move across the acoustic field. In another realization the array of transmitters and the array of receivers are arranged perpendicularly to each other. The transmitters are turned on one by one, so The authors are at the Computer and A u t o m a t i o n Institute, Hungarian Academy of Sciences, Budapest, Hungary. Paper received 2 May 1978.

ULTRASONICS. NOVEMBER 1978

(2.1)

Zp z

_~

Plane S Fig. 1

Diffraction pattern of a slit

0041-624X/78/060251--08 S02.00 © 1978 IPC BusinessPress

251

where K is a constant depending on the amplitude of the illumination and on the geometry, z is the distance between the plane and the slit, d is the width of the slit, x the coordinate in the diffraction plane and X the wavelength. The width of the slit can be found from the distance between two minima of the intensity Xz

d -

(2.2)

2 ( x i - Xi- 1) Sweeping the frequency causes the diffraction pattern to change: increasing the frequency the pattern compresses, decreasing the frequency (increasing the wavelength) the diffraction lines widen. Assuming that a detector is placed at a distance Xo from the centre of the diffraction pattern, it detects a signal similar to that if it were moving along the diffraction pattern of a wave of constant frequency, as shown in Fig. 2b, where the detector is placed at Xo, and the frequency, v, is increased, as compared to Fig. 2a, where the detector moves and the frequency is constant. Now the size of the slit can be expressed by the difference between the frequencies of two minima:

d =

z

_

2X0(1/Xi -- 1/~i- 1)

zc 2Xo(Vi - -

1)

In this expression v = c/X was used, where c is the velocity and v the frequency of the wave. These relations are used to determine the wavelength of the light knowing the size of the slit. The intensity variation experienced by a detector moving along the diffraction pattern with a constant velocity V is

K I(t) = -~

[

sin((27rd/zXo) Vt) (2nd/zXo) Vt

I

2 (2.4)

In our case the detector is fixed and the frequency varies with the sweeping rate m = Av/At. The detected intensity as a function of time will be

l(t)

= __ Kc mt

[sin((21rd/cz)xomt) ] 2 [

~

t

vc I~

a Fig. 2

(2.5)

XO

tP

~ IJ,

mxo

(2.6)

PO

The only difference is that the heights of the maxima increase with frequency, owing to the mt factor before the brackets in (2.5). It can be seen that the distances between the minima in the pattern are decreasing with increasing frequency, but the energy of the fringes is constant. This effect can be compensated for by decreasing the intensity of the illuminating wave in a suitable way. In the acoustic case the wave attenua tion increases approximately quadratically with frequency, so a natural partial compensation can be achieved in the case of frequency sweeping.

t

Fourier diffraction Assuming an object of transmission function t(x,y) in the front focal plane of a converging lens, we can express the diffraction pattern in the back focal plane as the Fourier transform of t(x,y) 7 +~

E(f~,¢~)

j Xz

t(x,y) exp { - j 27r(xft +Yf,) ] cLx "_oo

(3.1)

where x , y are the coordinates in the object plane, z is the focal length of the lens, _ ~v.

f~-

Xz

f~_

cz'

~ _ ~v Xz cz

(3.2)

are the spatial frequencies, ~ and r/are the coordinates in the plane of the diffraction pattern, and X is the wavelength (see Fig. 3). From this latest expression it can be seen that it is possible to change the detected spatial frequencies either by changing the coordinates in the image plane or by varying the frequency. The former case is one of the conventional synthetic aperture methods, where the detector scans across the image plane. The latter is a new principle, in which, as shown in the previous section, the diffraction pattern moves in the image plane. This variation is, of course, not detectable on the optical axis, where ~ and r7 are zero. Let a detector be located at (~o, 770). In the case of the frequency sweep the velocity of its virtual motion, (see Fig. 4a) because of the compression of the diffraction pattern (Fig. 4b), is

b

The detected signal: a--in the case of moving detector; b--in the case of altering frequency

252

V -

The effect of the frequency sweep on twodimensional diffraction patterns

(2.3) Vi-

From (2.4) and (2.5) it can be seen that the shapes of the detected signals are similar if the speed of the detector inthe case of constant frequency relates to the speed of frequency sweep as follows:

V~ = ~o . d r .

Vo

dt

Vn

7?o

dv

Vo

dt

(3.3)

ULTRASONICS. NOVEMBER 1978

are the initial spatial frequencies. As shown in Fig. 4 the expressions (3.6) mean that the detectors seem to move on the radii determined by the intersection of the optical axis and the image plane and by the loci of the detectors.

~q

Ay

Fraunhofer diffraction

l,~lo n e

v

When the distance z between the object and the image planes is large enough (Fig. 5) the diffraction pattern can be expressed by using the Fraunhofer's assumption.

Imoge plene z

A E(~,~) = j~ Fig. 3

exp(jkz+jkx2+y ~ 2)

x

Optical Fourier transformation +~

J 7 fix,y) exp(-jk(x8 + yr~)) dx dy

/Detector

[\.,1'

e'ec'od

(3.8)

Now we express (3.8) as a function of time for the cases of scanning detector and of timed detector with frequency sweep. For a moving detector it is

I

A E(r) = - - voexp [Jc~ u°(222 + ~2(T) + ~2(T))] ]cz (3.9)

fft(x,y)

Fig. 4 Virtual aperture synthesis in the case of: a--virtual motion of the detector; b--compression of the diffraction pattern

--~

exp

- j - - vo(x~(r)+yr~(r

dxdy

CZ

and for the frequency sweep +oo

Let du/dt = m be constant. If the scanning begins at t = 0, then the scanned frequencies can be expressed as follows x exp [ -j 2n v(r) (x~° +y % )

f~ =f~o +df~] "t dt ~o

(3.10)

dx dy

(3.4) fn

=/%+d~dt I

no

In the above expression r means the time and the detector is supposed to be placed at (Go, 7/o).

"t

where

The integrals in (3.9) and (3.10) yield the same value when

[ = Go d__v = m~o dt [ ~o dt cz

~(r) = "~ v(r) VO

(3.5) dfn I = 7o dv - moo dt no dt cz

r/O-) = r/° v(r).

(3.11)

PO

Thus we get

A = &

+

vV °

x

CZ

(3.6)

g

+ Y'Tt"°

= &

Cg

where fG

V

-

Vo~o; fno CZ

ULTRASONICS

-

(3.7)

Vo~o CZ

. NOVEMBER

1978

Fig. 5

L~l°ne

Jl

Fraunhofer diffraction

~fi3

This is the same condition as (3.3) in the previous case. The complex part of (3.8) before the integral also yields the same function for (3.9) and (3.10), multiplying (3.10) with the function of frequency

(1

Y(jw) =B__wexp j ~ (4~2w2-q~1w 2)

)

Pl

sl

(3.12) S

P I

where

q~o = 27r ZUo;¢a = Z (2z2+~+r12o);~2= L . C

CZ

CZ

/)0

(3.13) co = 27r/);Wo = 2rVo. s, o

The result of omitting this phase correction leads to spherical distortion, but this can be corrected at the reconstruction.

Fig. 6 Evaluating the mcf: a - - t h e original geometry; b--replacing S with the sources S 1 and S 2

The effect of frequency sweep on coherence requirements The Mutual Coherence Function (mcf) is a very good tool to analyse the coherence requirements of imaging systems. It bears a close relation to the intensity of light, but can be handled in the same way as the amplitude. According to Fig. 6a, where the light or sound waves originate from the source S, and travel through the points P1 and P2 and interfere at the point P, the mcf can be expressed as n 1

the different frequencies/)1 and/)2 respectively. These frequencies are

vl

=

v(t-

tl);

T

S1 C

v2 = v(t- tl + At); At

-

(4.3)

Sl-S2 C

(4.1)

With a linear approximation the phase difference between the waves originating from $1 and $2 is

-T

t-- t t +At

where xl and x2 are the position vectors of P~ and P2 respectively, E is the electromagnetic field, At is the delay arising from the difference in the optical paths, ~ denotes the complex conjugate, and t is the time. The normalized mcf is

3'(At) =

tl -

m/) = 1)2 - - / ) 1 "

rl,2(At) = P(xbx2,At) = lim - - x T-~= 2T

f f E(xb t + A t ) • E(x2, At)dt

b

P1,2(At)

(4.2)

A~b(t) = 27r

f

V(T)dr

t--Jr 1

21r

t-t,+Atr

[ t~ ti

Iv(t-t,)+ L

dr(t- t0(r_

]

t+tO dr(4.4)]

dt

Evaluating (4.4) we obtain

[ r l , , ( 0 ) • r=,2(0)] v,

where F~,x and P2,2 are the intensities at P~ and P2. In our case these expressions have no importance in their original form, because the diffraction pattern varies with time. In our case, referring to Fig. 6a, the frequency of the waves transmitted by the source S and travelling on the paths sl and s2 respectively, differs when they meet at P, because of the frequency sweep. The effect of this difference can be shown by replacing the geometry of Fig. 6a with that of Fig. 6b where the acoustic sources S~ and $2 radiate with

254

Ac~(t) = 2Try(t- tO" At + 7r --dP[

• At

(4.5)

dt I t - t I

In this expression the first part originates from the path difference, and is also observable in the case of stationary illumination. The second part depends on the rate of frequency sweep. The normalized mcf can be expressed using (4.1), (4.2) and (4.5).

ULTRASONICS. NOVEMBER 1978

W = [7~:[=lexp(JAqS) ' 2T

/

exp (j2nA~b)dt ]

~- T

~ 0

f

f

T~°°

f

(4.6) This means that the visibility, '~ W, is zero, because the interference pattern vibrates with frequency Au.

H

o

f

LI

To avoid this vibration the frequency must be changed in steps. P

Applications In this section a few examples of possible applications are shown in the fields of acoustical and optical holography or pattern-analysis. We note here that the obtainable ratio of the maximal and minimal frequencies in one process can be about 10 in acoustics and about 3 in optics. 8'9

Fig. 7

Optical correlation

that by changing the frequency it is possible to recognize figures (characters, for example) of different sizes. The scaling theorem of Fourier transform states that m

Aperture synthesis As was shown in the preceding sections, the diffraction pattern widens or compresses with the variation of the frequency of illumination. This means that a fixed detector registers a series of spatial frequencies, placed on the same radius. The length of this scanned portion depends on the rate of the final and the initial frequencies. The method can be applied in non-destructive testing (ndt) or in medical diagnosis, where the Fourier imaging can be easily realized. In these applications it is a great advantage that the system does not contain moving mechanical parts, and the required electronic processing can therefore be done in vivo or real time.

Optical pattern recognition Optical pattern recognition uses the cross-correlation property of the filtering of the Fourier spectra. This means that the Fourier transform of the cross-correlation between the functions g and h is the multiplication of the Fourier transform o f g and the complex conjugate of the Fourier transform of h: lO

~'-I g(x) , h(x + e) I = G(jfx) " #(Jfx) where - denotes the complex conjugate, • the convolution, G is the Fourier transform o f g and H is that of h, /~ means 'Fourier transform o f ,

fx-

e Xz'

(5.1)

X is the wavelength, and z the parameter of the geometry. The realization of pattern recognition is illustrated in Fig. 7, where O is the object, H is the hologram of the original pattern, L1 and L2 are lenses, and P is the correlation plane. When the object pattern contains figures similar to the original pattern, a sharp, intense spot appears in the correlation plane. The shortcoming of this method is that the figures in the tested object and those in the process of constructing the hologram must have the same direction and size, otherwise the recognition is not satisfactory. We show

ULTRASONICS. NOVEMBER 1978

where m is a constant, x is the coordinate in the object space,fx is the spatial frequency, and Tis the Fourier transform of the object function t. It means the smaller the object, the wider the spectrum. As was shown in the second section, this can be compensated for using light of smaller wavelength. It is easy to see that the recognition of characters of different size can be done by sweeping the frequency in discrete steps. The position of the self-correlation spot shows the location of the recognized character. The ratio of the wavelength used at the construction of the hologram to that at which the maximum correlation appears, gives the ratio of the size of a character in the original object to that in the tested object.

Results of computer simulation Since we have had no possibilities to perform acoustical experiments so far, we executed a computer modelling process. A program was written in assembly language for a small-size EC 1010 type computer, suitable for interactive applications, and was used to analyse the filtering of the Fourier spectrum of two-dimensional objects. This program was explained in detail in referenceJ a A 16 step linear grey scale was used to visualize the 64 x 64 elements of the array of objects, filters and spectra. The reference wave was usually omitted, because the computer could store the wavefront samples in complex form. A complex software arithmetic was written to evaluate the computations in simple word, 16 bit, floating point format. It was necessary to use this format because of storage limitations. The transformation process of an object of 64 × 64 samples lasts 80s. The original object is shown in Fig. 8. The figures which the object consists of were chosen so that they contain both rough and fine parts, similar to common objects. The following figures show the consequences of the detection with different arrays of detectors, modelling the case of frequency sweeping.

255

~|

l

iII

I. '

Ii

II I~

•. •

,

, ~I

II

|,

, III'

I~I III I I I I I• I IIiIl • 1"1 IIIIIIIIII~II Ill• IIIIIIl |IIIII IIl l~IIIllIIIIIl ~III I• IIi III IIII~ l~I I III I I• • IlllI I• il~I Ill IIIIII IIII~IIIIIIIII IIIIIIll IIIIIIIIIIl It~IIIIIl IIIIIIIII II|IIIIIIl IIIII IIIIIIIII iI~IIIIII I~I I~I~II Illl IUIII

IIl II I

I

,

•

I

•

I

I

IIii II II

II

II IIi|

~l

II •

fill

II

II

Ill

I •

I

II

iI

iI

I !

I

I

iII

I

ii

I

n I•IIl rill

IN

It e 1•11

m~ I e|

iIII

g•l •• | | I I ! ~g~ ill | | I I I ~ l l l ~'|11 n • • • II R gl ~ !1 i t / I I 1 ~

fig n

III

I

II

i

.,

!

t

:

It

,

itIin

,, III

•

i

,

l.,..

•

•

I

#.m.

I~

#

,

i

,o 1

I

I

!

II II I

||

I I I |

|I

IIIIIll! I •

-" II

|

I

HII

a Fig. 8

The original object

In Fig. 9a detectors placed around the optical axis with their 'run out' as the frequency of the sound varies are shown• The black dots represent the elements of the spectrum (that is, the equivalent detector matrix) which were taken into account in the reconstruction process. From the reconstructed image (Fig. 9b) it can be seen that the quality of the picture is bad, because of the arbitrary choice of the placing of the detectors. In Fig. 10a new detectors were added in the higher spatial frequency area, and the placement of the detectors was chosen more systematically. The reconstructed image (Fig. 10b) shows that the rough parts of the object can be recognized, and the noise has disappeared, but the fine details are smoothed, because of the relatively small amount of the high frequency components detected• In the case shown in Fig. 1 la 67 detectors (that is, the same quantity as one row) were placed. The quality of the reconstruction is now acceptable, as shown in Fig. 1 lb.

~

~ o0

From these results it is obvious that from a relatively small amount of data placed on the radii, the reconstruction can be carried out to an acceptable quality.

Summary It was shown that in the case of Fourier and Fraunhofer holograms aperture synthesis can be achieved by sweeping the frequency. Computer modelling showed that the information collected this way is enough for reconstruction. The method can be used in aperture synthesis, in optical pattern recognition and in the determination of the sizes of objects.

256

Fig. 9 Detection with 13 detectors: a--the detected spatial frequencies; b--the reconstructed image

ULTRASONICS.

NOVEMBER

1978

,

: ,

: ,

l II

I !

I I

~ I

iI

,

:

I I

.*,,

iI

~

I

~

!I I

i I

II

I I

I ill

I

., I I

Inlr•I

| I

iI

tI

I I

! I

II II

II

,, I!

I ~ I! II ii1t I I 1 1 1 li IIII I! I II I II !111 I11i It ttll II tllI IIIilliliiliiiI IIIIIItlIlilIIIIIIIIIIlIIIIIIIIHItlIilIillIIIIIIIIIIilllll Iiiiiiiiiiiiiil illl /i ltlil Ii Jill IIii il I tli ! ii IIII iill!lll I| I I l I t tl ii i I I i I II Illl Ii i 1 t I t II llIl II I II I! II tI 1 il I II I I I I I t I 1 I I ! I I I II I I i I ~ II I I l ~ I I li I It I | I I ~ I l ii t I • • I I I I I ! t I I I ii I t i I I I I lilt

II

iill

,

!

!1

111t

,

i

:

I

II

:

II

il

o_

ltll

IIit IIII

IIII

i111 IIII I!11

II It II

II IIII II

I • l I i I I|

',

:

!

,

II

II!1

I

!

II

I

:

: :

,

'.

,

',

:

b

a Fig. 10

"

II

!! |l

I I I11,l

I

!1

lI !

'

:,,:

!I

II

I =

l

°.

,'

:

".V

!

,t~=,

Ii

I I

,

...

,'

:

~ ! l ~

:.

,'

:

Detection with 37 detectors: a - t h e detected spatial frequencies; b--the reconstructed image

, I I

*

,,,

I

I I

I

I I

i

",

II

t, •

I

, I

I

|

I I I III I I I I

• I I I

l,,,,

i

:

I I I | I I

•

•

,

I I I I

I I I I

:

• •

,,,

:

I I

,", i .. !,I IfI',,,,•'I ,..• I',.,I"I ,.,,,, II I III I

,

I I I I I I I II I • I

:,'1

I I

I I I I I I

,

I I

:

I

, X*llg*

I

I

~

I ,

I

•

I

,'

•

I •IliIiiII lllllll

. ,, i ::..':-

I • II I I I II I I I II I I I I II II I I II I I I • • II I I II II II I I I I II I I I II I I • • II II ! I I I I! ! I I i • II I I I ! II I! ! I | III I I I Ili I I I II I1 !1 I! It I i !! t I I I I II I I II II tl II ii I I I I III I I I Ill I I I I II II II~ i |t ! I I Itl i i ! tll I I ! ~1 I I lIl tI ll li ! I t I!1 III III I I ! II !1 II IlI II i t~ I I II II! II I i II i II III II~l ~1 t~11 I I 1111111111| • I ! ! 1 1 I! i111 II lil~ II~ IiIl I I lllllil|l I I IIII III IIII li lllII IIII ~I I I I I t l l l t l l t l l l • I!1 II I ! 1 1 !1111 ~l|lllliI !~!111111 I I I II!111111 Illllllil llllIlIl illl~ I II i III!1 IIIIIIll IIIII!III!1 I II! illllllllll II Illllllllllllllil lilt Iitiillill lll~lllllllllllllllllllllllllll IIi111111 lIllllllt IIIllllllllllllllll IIIIliillli i1111 IIItlllllli 11111111 I I I l l I Ill I Illll III!1111 IIIIIII11 lilllllll I I I Illllilll I!1111111 !!111 III! II III I IIiiiiiiiii !IlI II !!11 IIII! II !111 II! IIII I I Iit111111 I t IIII I!1 IIII II tllt II IIII I I lllllllllll I I llll II I1~1 III I! I !1 I I II I11 tl I I II I II III !1 II !1 1 I I Ill III III I I ! II II II II! I II I I I tll t t I !11 I I I II II I11 II • I I I III • i I llI I • i ! II II It II Ii I I ~1 I • I I I I I • I I! II II II i' ~ II II Ill i I I I'll 'I I 'l ilI II II I I!i IIII I I III t I I II I • • I II tl ll ii I il I I i I I tl • I II Illl • II I • I I I !1 I I I II I II

..'

II I

:

I

:

•, I

I

: II

I

I

:

.

Il !

! ....

,, I

-

III

I

, ,.::

I I I III

• !

:: II

•

I • i I I I I ! • I I I

111

: : :

:

:

=..•. •,•,, =o

I ~

:I

,

I I !

I I I I

:

,,"

:1',

I I

I I I !11

• !

:: I

: I I I

i I • |

:

"

,. ! I : I I

I I I I I

11:

,

I!

II

" •

I

:

•

•

. XlI UMIIII -=

"*

#

" II

, It

• II

I

:

I

',

a Fig. 11

Detection with 67 detectors: a--the detected spatial frequencies; b--the reconstructed image

ULTRASONICS. NOVEMBER 1978

257

References 1 2 3 4 5 6

258

Metherell, A.F. 'The Present Status of Acoustical Holography' Seminar Proc Developments in Holography Boston 1971 137 Mueiler, R.K. 'Acoustic Holography' Proc IEEE 59 (9) (Sept 1971) 1319-1335 Harmuth, H.F. 'Transmission of Information by Orthogonal Functions' Springer-Verlag Co (1970) Berlin Pratt, W.K., Kane, J., Andrews, H.C. 'Hadamard Transform Image Coding' Proc IEEE 57 (1) (Jan 1969) 58-68 Metherell, A.F. 'The Relative Importance of Phase and Amplitude in Acoustical Holography' Acoustical Holography 1 Plenum Press New York (1969) 203-221 Berbek/tr, Gy., Kazsoki, J. 'Optikai Eszkozok K6szit~se Sz/tmit6g6ppel' BME Budapest 1974

7 8 9 10 11 12 13

Goodman, J.W. 'Introduction to Fourier Optics' McGraw-Hill Book Co New York (1968) Brown, A.F., Weight, J.P. 'Generation and Reception of Wideband Ultrasound' Ultrasonics 12 (July 1974) 161-167 Brignall, N. et al 'Tunable Far Infrared Generator by Difference Frequency Mixing in InSb' Optics Communications 12 (Sept 1974) 17-20 Champeney, D.C. 'Fourier Transforms and their Physical Applications' Academic Press London (1973) DeVelis, J.B., Reynolds, G.O. 'Theory and Applications of Holography' Addison-Wesley PCo Reading (1967) BerbekAr, Gy. 'Hologramapertfir~k Letapogatdsa a Frekvencia V~iltoztat~is~ival' BME Budapest 1975 Berbek~, Gy. 'Computer Modelling of Holographic Imaging' Preprints of Int Conf on Optical Computing 1977 Visegrhd, 27-38

ULTRASONICS. NOVEMBER 1978