Signal Processing 4 (1982) 249-262 249 North-Holland Publishing Company

SUBSURFACE RADAR SIGNAL DECONVOLUTION*

Isabelle PAYAN, member EURASIP Thomson CSF, Etude et D&:eloppement d'imagerie nou~'elle, 48, r,w Camille des Moulins, 1='-92130 [ssy-les-Moulinea,~x, France, (formerly with: Uni~'ersity of Southern California, Medical Imaging Science Group, 4676 Admiralty Way, Suite 932, Marina del Rey, CA 90291, USA)

Murat KUNT, member EURASIP Laboratoire de traitement des signaux de I'EPFL, 16, chemin de Bellerive, CH-I007 Lausanne, Switzerland

Werner FREI Un&,ersity of Southern California, Medical Imaging Science Group, 4676 Admiralty Way, Suite 932. Marina del Rey, CA 90291, USA

Received 24 March 1981 Revised 2 November 1981

Abstract. We present two methods of signal deconvolution for systems whose impulse response (wavelet function) can be explicitly determined, and where the goal is to locate short impulses in the presence of strong, reverberation-like interferences.

The first method, which we call algebraic deconvolution, differs from other known techniques in two ways: first of all, explicit use of the wavelet function provides more powerful a priori knowledge than the autocorrelation or the power spectrum. Secondly, this method permits to flexibly trade off noise versus resolution.

In the second method presented here, we use an analytical model (synthetic wavelet) of the system impulse response to determine an inverse filter.

These methods have been developed for video pulse radar signals, and encouraging early results have been obtained.

Zusammenfassung. Zwei Methoden der Entfaltung von Signalen werden beschrieben ftir den Fall, da~3 die Impulsantwort des verwendeten Systems explizit angegeben werden kann. Ziel der Untersuchung ist hierbei, kurze Signalimpulse in Gegenwart starker nachhallartiger Interferenzen ausfindig zu machen. Das erste Verfahren, bier als 'algebraische Entfaltung' bezeichnet, unterscheidet sich von anderen bekannten Verfahren auf zweierlei Weise. Dadurch, da~3 die Impulsantwort explizit bekannt ist, erh~ilt der Benutzer yon vorn herein eine sehr viel wirksamere Information also bei Verwendung der Autokorrelationsfunktion oder des Leistungsspektrums. Des weiteren gelingt es mit diesem Verfahren, zwischen der Rauschempfindlichkeit einerseits und dem Aufl6sungsverm6gen andererseits einen tragbaren Kompromi~ herzustellen. Beim zweiten Verfahren wird ein analytisches Modell, d.h. eine synthetisierte Impulsantwort dazu verwendet, ein inverses Filter zu entwerfen. Die Verfahren wurden ffir Videoimpulsradar entwickelt. Die ersten Ergebnisse sind sehr ermutigend.

R6sum6. Nous pr6sentons ici deux m6thodes de d6convolution de signaux produits par des syst~mes dont la r6ponse impulsionnelle (onde) peut ~tre d6termin6e de mani6re explicite. Le but poursuivi est de Iocaliser de courtes impulsions m~16es h d'importants parasites provenant en particulier de r6flexions multiples.

La d6convolution alg6brique diff~re des m&hodes classiquespar deux aspects: tout d'abord elle utilise la connaissance explicite de ta r6ponse impulsionnelle, ce qui fournit plus d'information que son autocorr~lation ou son spectre; enfin, le compromis entre la r6duction du bruit et la finesse de la r~solution peut se traiter tr~s souplement par I'introduction d'un facteur d'6talement.

* This work was performed under the auspices of the U.S. Department of Energy Contract No. DE-AC08-76NV01183 through a subcontract from EG&G's Santa Barbara Oper- ations.

Reference to a company or product name does not imply approval or recommendation of the product by the U.S. Department of Energy to the exclusion of others that may be suitable.

0 1 6 5 - 1 6 8 4 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 O 1 9 8 2 N o r t h - H o l l a n d

250 L Payan et al. / Subsurface radar signal deconL'olution

Dans la deuxi~me m&hode present,e, nous utilisons un filtre inverse qui est d&ermin~ par un module analytique (onde synthehique) de la r~ponse impulsionnelle.

Ces m~thodes ont ~t6 d~velopp~es pour des signaux radar h impulsion video, et les premiers ff:sultats sont tr~s encourage- ants.

Keywords. Video pulse radar, deconvolution, inverse filter.

1. Introduction

Deconvolution designates the recovery of a

signal degraded by the response of a linear system. Examples of such degradations are image blur caused by optics and atmospheric turbulence [1], acoustic signals obscured by multiple reverberations [2], and more generally all measurements made with systems which exhibit a non-uniform frequency response. The recovery

of the original signal, or, more precisely, the esti-

mation of this original signal presupposes some

knowledge about the characteristics of the signal,

the degrading system response and the noise

respectively. The existing techniques thus center around the kind of a priori knowledge obtainable by measurements or reasonable assumptions.

Wiener filtering [3], for example, presupposes

knowledge of the signal and noise power spectra, as well as the system response, to specify a filter

which minimizes the mean square error between the original signal and the estimate obtained.

Linear prediction, on the other hand [4, 5], utilizes the autocorrelation of the degraded signal to predict its future behaviour, assuming that pre- dictability is caused by the response of the degrad- ing system. The desired signal is then extracted by measuring deviations from the predicted values.

To cite one more example, complex cepstrum filtering [5, 6] presupposes that the measured signal can be mapped into a 'quefrency' domain, wherein convolved functions occupy different regions and can thus be separated from each other.

In comparison, both methods presented here presuppose an explicit knowledge of the system impulse response to calculate the coefficients of the deconvolution filter. In the presence of noise, Signal Processing

both filters can be adjusted to trade off resolution, e.g. the ability to discriminate closely spaced

events, with the ability to detect weak signals

otherwise obscured by noise. These methods were developed for video-pulse

radar [7], which utilizes pulses of very wide band- width to penetrate media with high absorption losses such as ground or water. This is done to detect the presence of buried objects, void near mine shafts and so forth, which present dielectric

discontinuities and thus cause a part of the incident

electromagnetic energy to be reflected I~ack to the receiving antenna. Hence the received signal

would ideally consist of pulses indicating the pre- sence and range of such discontinuities. In reality however, these returns have the appearance of damped oscillations which often completely obscure the desired information. Fig. l(a) shows an example obtained by moving the antenna of such a radar across the side of a gravel pile. The

heavy slanting striations visible in this image are caused by the interface between the bottom of the

gravel pile and the ground. Our aim here is to reduce these striations to a single line indicating

the presence of the interfaces.

2. Mathematical model

2.1. Situation of the problem

In our approach, we assume that a measured radar signal is the convolution between the ideal radar returns (consisting of a random train of pulses) and a 'ringing' or wavelet function, which is in fact the system impulse response (see Fig. 2). In the following, the 'wavelet' designates this par-

ticular function.

L Payan et al. / Subsurface radar signal decon~'olution 251

?eDtZ -

rdTrio~ CD~:: [ ~ o a d . d e o im z r ~ v e l ]

F ~ Z ~ [ j ~ C V~22F e _ ~ . . . . . .

T ~ . 20FVQS2]~Z ~0 B sm~!z21 r -3~- ~nsD4n%eFe3

^hi~a recorsins

life

2 5 5

0

V~ V~Vl ' v ~ Y ~ - , ~ , ~

V Fig. hb). One scan line.

Vot. 4, No. 4, July 1982

252 L Pavan e" al. / Subsu~.,'ace radar signal deconcolution

Fig. 2!a). Random train of pulses.

Fig. 2(b). Ringing function.

AAAA_ VvV V

Fig, 2(c). Convolution of 2ta) and 2ib).

First we need to extract the wavelet f rom the

received radar signal. This operation seems very

simple, for example when we look at the signal

obtained by moving the antenna of the radar over

a swimming-pool filled with water (Fig. 3(a) rep-

resents the image, and Fig. 3(b) shows one scan

line where we see the "ringing' function caused by the water /concrete boundary return). Notice that

the wavelet is not always so clearly visible (see

Fig. l(b)).

Once the wavelet is known, we can perform an

"inverse convolution', or deconvolution, of the received signals to recover the actual signals as

closely as possible. This will simplify the given images (composed of several consecutive signals)

since the deconvolution filtering transforms the wavelet into the desired function. This desired function is in fact the ideal radar return that we would obtain from a boundary if there were no echo effect. Such an ideal return would be a Dirac

Signa~ Processing

pulse locating the boundary at a unique depth.

For reasons explained later, we shall also consider

smoother desired functions such as a gaussian

curve ,

2.2. Formulation of the problem

From a mathematical point of view, the formu-

lation of our problem is as follows:

Given the wavelet w(0) . . . . . w(N - 1)

find a filter f(0) . . . . . / ( M - l) ( M > N )

such that the convolution (P1)

c(n) = w(n )* f (n )

Vn = 0 . . . . . M + N - 2

be as close as possible to a desired

function

d(O) . . . . . d ( M + N - 2).

[. Payan er aL / Subsurface radar signal deconvolu&~n • ""

depth ~Z

rs:!a:<ians c:

:~ez~! bars

etal bar~

~axi~w~ decs~ in ~he ima~

3 ~ r

Fig. 3tab Original image of swimming-pool data.

In the next two sections, we present two solu- tions to this problem. One defines the convolu- tion and the error measure in the time domain. The second solves the problem in the frequency domain; it uses a modified version of the inverse

filter method.

3. Algebraic method of deconvolution

3.1 . E x t r a c t i o n of" the w a v e l e t

The swimming-pool data is very attractive because the wavelet is clearly visible (see Fig.

3(b)). Indeed there exists a unique boundary VoI. 4, No. 4, July 1982

254 L Payan et al. / Subsurface radar signal deconvolution

255

/ - "-V

Fig. 3(b). One scan line.

water /concrete in most of this image. The wavelet (see Fig. 4(a)) is extracted from an average of several signals obtained from scan lines over the

swimming-pool. The averaging permits to know

the wavelet with a good approximation. We assume that the shape of the wavelet is not

modified from one environment to another; only its frequency, or its period, might differ. Thus for different data we may need to either adjust the

filter to the data, or adjust the data to the filter.

This adjustment factor can be determined very

easily. We mentioned in Section 2 that a measured radar signal is the convolution between a random

train of pulses and the wavelet. A random train of pulses has a uniform spectrum. The wavelet is

a deterministic function and its spectrum presents one or several maxima. Consequently the maxima found in the received radar signal are exactly the

same as the ones of the wavelet. If they are not, an adjustment is needed so that they match.

Signal Processing

]

Fig. 4(at. True wavelet.

I. Pavan e; al. / Subsurface radar signal deconco!,mon 255

Fig. 4 ,b) . S y n t h e t i c w a v e l e t .

3.2. Definition of the algebraic filter

The algebraic deconvolution filter is the solution

to problem (P1) stated in Section 2, whereby we chose the least square criterion to measure the

error between the desired function d(n) and the

convolution c(n). After developing and simplify-

ing the equations (for a matter of readibility all

the details are explained in Appendix A) problem

(P1) is reduced to the following linear system of

equations:

(P2)

Given the wavelet w(O) . . . . . w(N - 1),

given a desired response

d(O) . . . . . d ( M + N - 2 ) ,

find the filter f(0) . . . . . f ( M - 1) such that

U . F = D

where:

(1) U is the matrix displayed in the adjacent column.

(2) F = [ f ( O ) . , . f ( M - 1)]'.

(3) D = [Ra~.(0) • • • Ra,,.(M - 1)] I.

.%'-1

(4) R ~ I i ) = ~v win+i).w(n) n =0

is the autocorrelation of the wavelet function w.

,%" - 1

(5) Rd~,.(j) = V d ( n + j ) ' w ( n ) n = 0

is the crosscorrelation of the desired response d with the wavelet w.

U =

I R (0~ ".. R ~.V-I~ 0 . . - l)

I R ~ i N - I} " ' - - . " ' - . 0

L 0 "-- 0 R ~ ' , \ ' - I p " . R~qH,

Since U is a symmetric definite positive, and

hence nonsingular matrix, the problem (P2) has a unique solution which can be stated as follows, according to the definitions given above:

F = U - ~ . D . \ o l ~, Na. 4 Jub 19S"

256

.~.,~. The results

The first choice of a desired response is a Dirac function:

d ! n i = l i f n = } < X l + N - 2 ) ,

d(n ! = 0 otherwise.

In the filtered signal, we expect a large value

each time the wavelet occurs in the received radar signal. The result obtained with the swimming-

pool data isee Fig. 5ib}) is satisfying, although two

drawbacks can be observed. A high peak is sur- rounded by two low peaks; one reason may be that we use a finite length filter. Second drawback:

the noise increases with the range. This is because the radar receiver automatically increases gain as a function of range to compensate for the high absorption losses in the media of interest. The

deconvotutiorr filter is of a high-pass type, however, and tends to further enhance the noise encountered at larger ranges/see Fig. l(b)).

Therefore, to improve the results, our next

choice of a desired response is a Gaussian function:

d ( n ) = e x p [ - ~ ( , n l -n , , I : ]

[. Payan et a'. Sz+bs +r/a,:'~ radar 9i~*',a" deconLc~e~ticm

where no=½, M + N - 2 ! and a is a parameter which characterizes the width of the function. This

choice has the effect of reducing the high frequency gain of the deconvolution filter.

Figs. 6 and 7 show a real improvement. The

metal bars embedded in the concrete of the swim- ruing-pool can be easily detected. The result on

the gravel pit data, although not completely satis- factory yet, enhances the boundaries that are to be pointed out.

4. Inverse filter method-- ,nthetic wavelet

4. i. Introducson

The algebraic filter is obtained in two steps: first extract the wavelet from the given data, then solve

a linear system of equations. In this second method, we try to simplify these operations as much as we can. We create an analytical model of the wavelet, which we call synthetic wavelet by opposition with the true wavelet extracted from

the data. Then we formulate problem (P1) in the Fourier domain so that the convolution becomes

a multiplication [8]. The solution has indeed a very simple form.

b

a

|

° : + _

I "+

' . ° 2

Fig. 5(aL Filtered swimming-pool data using the algebraic method !a Dirac function is the desired responsei.

Signal P rocess ing

[. Pavan et al. Subsurface radar ~ignaI decont'oh~rion 257

2~5

i v ,

Fig. 5ibi. One scan line.

,!.2. The inverse fiIter

Consider the problem ~P1) stated in Section 2. The filter is defined by the equation:

wfn)*fin)=din) Vn = 0 . . . . M - , N - 2

C c n ~ 2 s

which, in the Fourier domain, becomes:

W(~o).F(w) = D ( ~ o ) Wo

where W, F and D are the Fourier transforms of the wavelet w, the filter/" and the desired response

J

4"

2, ̧

1

Studied scan line

Fig. 6, a!. Filtered swimming-pool data usin_o the algebraic method ~a Gaussian function is the desired response!.

\roI. 4, N~'. 4, JUI~ 1982

.=-~8 [. Pavan e" ~z[. S~b>'.zrt~zce radar signa[ dec,ont,:,[,cion

25

v" iv u v VV",~ fV'VWV" ̂ AAAA, I ' wwVVVl

Fig. 6~bi. One scan line.

d respect ively . The immed ia t e so lu t ion is:

FtoJ ) = D{co ~ / W!~o l.

Let us r e m a r k that. if d is a Di rac funct ion, D is

the cons tan t l , hence the name "inverse filter ' .

4 .3 . T h e s y n t h e t i c w a u e l e t

If the Fou r i e r t r ans form W of the wavele t is

known exact ly , then the filter is known exact ly in

the f requency domain , according to the above

equa t ion . That leads us to in t roduce the synthe t ic

l . : i , ' : a j . , ,'

t

[ t # '

i • ¢ f" ~ •

2 • t

. , 1 / i t , , ; '

" t i l ~ t ,, l

i l l '

Fig. 7(at. Filtered gravel pit data using the algebraic method ta Gaussian function is the desired response ~.

S i g n a E P r o c e s ~ m g

L Payan eta/./Subsurface radar signal deconvolution 259

2-25

LAt, 'Vrv't

IAllnl l V, I1

Fig. 7(b). One scan line.

wavelet (see Fig. 4ib)), which is an analytical model of the true wavelet extracted from the received radar signals; its equation is the fol- lowing:

w ( t ) = t: exp(-at)sin(2"rrfot)

where f0 is the maximum value in the spectrum of the true wavelet; a is a parameter chosen such that

t z exp(-at)[,=s,.2~o = e ;

5/2/o corresponds to the wavelet length, and e is the approximation of the zero value.

The analytic Fourier transform [9] of the syn- thetic wavelet is given by

2 B ( A z - B 2) W(ro) = ( A 2 + B 2 ) 3

where

A = a + iw, B = 2"rrfo.

4.4. The results

The only desired function chosen here is a Gaussian function, as described for the algebraic method in Subsection 3.3. Fig. 8 is the first result obtained. It tells us that we need to determine

more precisely the parameters of the synthetic wavelet as well as those of the filter.

5. Conclusion

The two methods presented here use very different ways to obtain the filter. While the first one extracts the wavelet from the received radar signals, the other creates a model of the system

impulse response. In one case the filter parameters are the solution of a linear system of equations; in the other, the filter is computed analytically

through an integration. For both methods, filtering can be performed in either the time domain or the frequency domain with the same ease; the criterion of this choice could be the signal or the filter length for example.

The advantage of the algebraic method of deconvolution is that it utilizes the true wavelet; but unfortunately it is not always possible to extract it, for example when the environment is non-homogeneous such as in the gravel pit data (see Fig. l(b)).

The advantage of the inverse filter method is that the filter is known analytically, thus exactly, in the frequency domain.

V o t 4, N o 4, July 1982

260 I. Pavan et a/. " Sub~ur.face radar signal deconL'o!zvion

Signal Processing

¢

Fig. 8(a!. Filtered swimming-pool data using the inverse filter method.

n

Fig. g~b~. One scanline.

I. Payan et al. / Subsurface radar signal deconvolution

The early results shown here are very encourag-

ing. We hope to be able to study more deeply the problem of parameters adjustment to obtain better results.

Acknowledgements (P1.1)

The authors wish to thank Mr. Richard Lynn from E G & G Inc. in Santa Barbara who is respon-

sible for initiating this project. This work has been

accomplished at the Medical Imaging Science Group (University of Southern California) and has

been possible thanks to Yves Fontaine who pro- cessed the photographs, and every person in the

group for their suggestions and help of any kind. Thanks are also due to anonymous reviewers who

helped us to improve our paper.

Appendix A

Mathematical model: details

This appendix explains in details how problem (P2) stated in Section 3 is derived from problem (P1). In the following, we assume that a signal s defined between indices NO and N1 (N0<N1) is equal to zero outside these limits, that is

s ( n ) = 0 i f n < N 0 or n > N 1

The initial problem (P1) is stated in Section 2 as:

(P1)

Given the wavelet w(0) . . . . . w ( N - 1)

find a fil ter/(0) . . . . . / ( M - l ) ( M > N )

such that the convolution

c(n) = w(n)*f (n)

Vn = 0 . . . . . M + N - 2

be as close as possible to a desired

function

d(O) . . . . . d ( M + N - 2 ) .

The chosen measure being the least criterion, the problem (P1) thus becomes:

square

261

Given the wavelet w(0) . . . . . w i N - 1)

and a desired response

d(O) . . . . . d ( M + N - 2 ) , find the filter

f(O) . . . . . f ( M - 1) that minimizes

,'*f ~ N - 2

E = Z [c (n ) -d (n)] 2

where c(n)= w(n)*f (n)

Vn = 0 . . . . . M + N - 2 .

The filter f minimizes the error E if and only if the partial derivatives of E with respect to every filter parameter f (m) is equal to zero. That is

OE - - = 0 Vm = 0 . . . . . M - 1 . Of(m)

By definition of the convolution

A I - I

c(n)= Z w ( n - k ) ' f ( k ) k = 0

Vn = 0 . . . . . M + N - 2

(1)

Therefore

Of(m) 2 ,X'=o ~-o3" [ w ( n - k ) . f ( k ) ] d(n)

M-, of( i ) ] ._- [ w (" -,") "~f-F~ ~j }

1 bE u - t M+N-2 Z }~ w(n-k) , w ( n - m ) "f(k)

2 Of(m) k:o , : o

M ÷ N - 2

- ~ d ( n ) . w ( n - m ) (2) , '2=0

We easily recognize the autocorrelation of the wavelet w :

N - 1

R~.,(j)= Y n = 0

w(n). w(n - j ) = Rw,, ( - D

and the cross-correlation of the desired response d with the wavelet w :

.~4-- 1

Raw(f) = Y~ d ( n ) ' w ( n - j ) . rl = O

VOL 4. NO. 4, Jul) [~8~

262

Thus the eq. (2) can be rewr i t t en as:

1 dE M-1 R~,w(k - m ) . f ( k ) - Rd~(rn )

2 Of(m) = k=O

C o m b i n i n g eqs. (1) and (2), p r o b l e m

becomes :

G iven the wavele t w(0) . . . . . w ( N - 1)

and a des i r ed response

d(O) . . . . . d ( M + N - 2), find the filter (P l . 2 )

f(O) . . . . . f ( M - 1 ) such that

M - 1

Z R w w ( k - m ) ' f ( k ) = ' R a , ~ ( r n ) k = O

'qm = 0 . . . . . M - 1 .

It is also poss ib le to use a f o r m u l a t i o n of this

p r o b l e m with matr ices . Then we get p r o b l e m (P2):

G i v e n the wavele t w(0) . . . . . w ( N - 1),

given a des i r ed response d(0) . . . . .

(P2) d ( M + N - 2 ) , find the filter

f ( 0 ) , . . . , f ( M - 1) such that

U . F = D

where :

(1) U =

I R.(0) . . R . ,~ ' -~ ) 0 ... o. ]

R ~ ( N - 11 0

' 1117-----_%"-" 0 ' ' ' 0 R'~.~{N - l ) ' ' ' " R (0)

(2) F = [ f ( O ) . . . f ( M - 1 ) ] ' .

L Payan et al. / Subsurface radar signal deconvolution

(3) D = IRa, ,(0) • • • R a , ( M - 1)] t.

N - 1

(2) (4) Rw, , ( j )= ~ w(n +i) . w(n) n=O

(P1.1) is the a u toc o r r e l a t i on of the wavele t funct ion w.

N - I

(5) Rd, , ( . / )= ,'7. d(n +/)" w(n) n = 0

is the c rosscor re la t ion of the des i r ed response d

with the wave le t w.

N.B. Figs. 5(a), 6(a) and 7(a) are ro ta t ed 90 °

coun te rc lockwise from their no rma l pos i t ion .

Reterences

[1] B.L. McGlamery, "Restoration of turbulence-degraded images", J. Opt. Soc. Am., Vol. 57, March 1967, pp. 293-297.

[2] T. Stockham, T. Cannon and R. Ingebretsen, "'Blind deconvolution through digital signal processing", Proc. IEEE, Vol. 63, April 1975, pp. 678-692.

[3] K. Kondo, Y. Ichioka and T. Suzuki, "'Image restoration by Wiener filtering in the presence of signal dependent noise", Applied Optics, Vol. 16, Sept. 1977, pp. 2554- 2558.

[4] K. Peacock and S. Treitel, "'Predictive deconvolution: theory and practice", Geophysics, Vol. 34, April 1969, pp. 155-169.

[5] L.C. Wood and S. Treitel, "'Seismic signal processing", Proc. IEEE, Vol. 63, April 1975, pp. 649--661.

[6] T. Ulrych, "'Application of homomorphic deconvolution to seismology", Geophysics, Vol. 36, August 1971, pp. 650-660.

[7] D.L. Moffatt and R.J. Puskar, "A subsurface electro- magnetic pulse radar", Geophysics, Vol. 41, June 1976, pp. 506-518.

[8] M. Kunt, Traitd d'Electricitd, Vol. XX: Traitement Numdrique des Signaux, Editions Georgi, Lausanne, 1980.

[9] G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York, 1968.