Lateral inverse filtering of ultrasonic B-scan images

LATERAL INVERSE FILTERING

OF ULTRASONIC B-SCAN IMAGES

H. Schomberg,

and G. Mahnke

W. Vollmann

Philips Forschungslaboratorium D-2000 Hamburg

Hamburg 54, F.R.G.

The formation of ultrasonic B-scan images using parallel beams may be modelled as a lateral, one-dimensional convolution of the beam profile and an unknown but wanted reflection coefficient. Lateral inverse filtering, or deconvolution, might therefore be used to improve the image quality. Two different deconvolution techniques are applied to both an image of a tissue mimicking phantom and a human liver. An enhancement of the resolution (defined as the reciprocal of the half-width of the image of a point reflector) of about 1.4 is achieved. This is in good agreement with the previously derived formula R = dm, which relates the signal-to-noise ratio, SNR, to the resolution enhancement, R. However, each method also creates artifacts, and despite the slight resolution enhancement, the deconvolved liver images do not exhibit more information nor are they more appealing. So it is felt that the computational effort is wasted. This failure is not a fault of the special deconvolution techniques tried here, but rather caused by the logarithmic dependence of R on SNR and by the noise level, which is largely due to macro- and microscopic inhomogeneities of the tissue and cannot be made arbitrarily small. Key words:

I.

B-scan image; verse filtering; signal-to-noise

deconvolution; Kaczmarz filtering; lateral inpoint-spread function; resolution enhancement; ratio; ultrasound; Wiener filtering.

Introduction

Ultrasonic B-scan techniques have become a widely used medical imaging modality [l-3]. The desire to improve the image quality has stimulated research not only in the field of ultrasonic measuring equipment but also in the field of digital data processing. In particular, several variants of Usually studies have been digital inverse filtering have been investigated. made on how such techniques influence simulated B-scan images or real images of overidealized objects, e.g. wires in a water tank [b-7], whereas tests on clinical B-scan images apparently have not yet been reported. It is therefore unclear whether or not inverse filtering can improve In this paper we attempt to answer this question clinical B-scan images. for the case of lateral inverse filtering of B-scan images. We shall conclude that it may not be worthwhile to do such filtering. A typical B-scan liptical shape, where sonic beam) dominates inverse filtering is under the assumption this calculation can the process of image system with parallel x and y as axial and 0161-7346/83 Copyright All rights

image of a point reflector has approximately an elthe lateral dimension (perpendicular to the ultrathe axial dimension (parallel to the beam). Lateral a method which tries to calculate an improved image that there is only lateral point-spreading. Before be carried out, it is necessary to derive a model of formation and degradation. To this end we assume a lines-of-sight and introduce rectangular coordinates lateral coordinates (see Fig. 1). Now consider a sin-

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0 1983 by Academic Press, Inc. in any form reserved.

of reproduction

38

LATERAL INVERSE FILTERING

gle image row corresponding measured image intensity dinate y. This function, lution of what might be chosen row, r(y), and a row, f(y):

to an arbitrary but constant depth x0. The along this row is a function of the lateral cooris now written as a one-dimensional convom(y)', called the true reflection coefficient in the lateral point-spread function associated with that

m(y) The noise account. ferent

function

n(y)

An equation of the rows may require

=

fly-y')r(y')dy'

s

takes

the

+ n(y)

experimental

form (1) different

is valid functions

.

(1)

and modelling

also for the m(y), f(y),

other r(y)

errors

into

rows, but and n(y).

dif-

Lateral inverse filtering is now defined as any method that tries to reconstruct the reflection coefficient r(y) in a given row from the measured intensity m(y) in that row, given f(y) in that row and using Eq. (1). A complete, two-dimensional image would be reconstructed row-wise. f(y), which describes the intensity The lateral point-spread function, varies only slowly with the depth profile perpendicular to the beam axis, x0 of the row. With an unfocussed ultrasonic beam, as it was used for this study, all rows of the image have approximately the same point-spread function. In this paper we will therefore assume that the point-spread function is depth independent. This assumption is not essential and can be removed if it seems to be inadequate. spatial

The deconvolution domain.

Frequency such technique

Fig.

1

1

can be performed

domain techniques has been used in

in the

frequency

make use of the Fourier [4]. The images underlying

domain

transform. One [4] have been

The axial coordinate x is related to the echo travel x=&/2 where c is the average sound velocity for the lateral coordinate y is determined by the transducer

A symbol like m(y) either or the value this function

denotes attains

or in the

time t by object. The position.

the function m of the variable y, at y, depending on the context.

39

SCHOMBERG ET AL.

obtained from wires in a water tank. The empirical result of [4] is that the resolution of the images (defined as the reciprocal of the lateral full width at the half-maximum of the image of a point reflector) can be enhanced by a factor of between 1.2 and 1.9. This result is consistent with a theoretical result of [81, which relates the achievable factor of resolution enhancement, R, with the signal-to-noise ratio of an image, SNR, by R=dw

.

(2‘)

The resolution enhancement is defined as the factor by which the width of the image of a point reflector is decreased. For the precise definition of SNR we refer to [8]. We just mention that SNR is defined in the frequency domain. Relation (2) has been derived for the deconvolution technique used in [k] and for the case of a Gaussian point-spread function. However, one may expect that a Gaussian shape is not too bad an approximation and that all filtering techniques have comparable upper limits for the resolution enhancement. The signal-to-noise ratio of clinical B-scan images is of the order of 50. For a signal-to-noise ratio of 50, relation (2) allows a resolution enhancement of 2. The signal-to-noise ratio and thus the resolution enhancement increases as the noise level decreases. By our definition of SNR in the frequency domain, SNR decreases - at a given noise level - for a decreasing width of the point-spread function. Thus, the narrower the pointspread function, i.e. the better the original image, the smaller is the expected resolution enhancement. 10

to

In this paper, we shall employ a Fourier may be seen as a simplified Wiener filter.

transform

technique

which

As an example for a space domain technique we describe how to estimate r(y) from a discrete version of (1) using a numerical method due to Kaczmarz. This approach is less well known, but appears also attractive. To find the best among the mentioned strategies we tested them on two B-scan images. One of these images was obtained from wires embedded in a tissue mimicking material, while the other one shows a human liver. Based on the test results, we discuss the merits of the various approaches and of lateral inverse filtering in general. We shall not find an appreciable advantage. II.

Formation

of the

B-scan

Images

The two mentioned B-scan images were kindly provided by Philips Ultrasound, Inc., Santa Ana, CA. They were made with a 3.5 MHz unfocussed, linear array consisting of 64 transducer elements. The array has a length of 16 cm. Each ultrasonic beam is generated by a simultaneous excitation of either 3 or 4 neighbouring transducer elements. An alternating change between 3 and 4 elements produces 123 equidistant ultrasonic beams. This change causes intensity differences between neighbouring lines of sight. The differences are so small that they do not disturb the appearance of the image. The 123 columns of the image add up to an image width of 15.25cm. The number of rows is 320. The electronic system, briefly, includes demodulation, time compensated gain, logarithmic amplification and analog-todigital conversion. The output of the system is a non-negative S-bit number for each of the 123 x 320 picture elements. The linear array used is an early, experimental model with a broad point-spread function and a correspondingly poor image quality. Nevertheless, the resulting images are good enough to substantiate the claim that

40

LATERAL INVERSE FILTERING

Fig.

2

B-scan image of a tissue originate from wires.

lateral inverse filtering cussed in the introduction hanced.

mimicking

is of little - the better

phantom.

The white

streaks

benefit, because - as it was disan image, the less it may be en-

The first B-scan image is presented in figure 2. It shows a phantom which simulates the reflection properties of human tissue. Inside the phantom there is a configuration of wires which appear as white streaks in the image. For our quantitative analysis of inverse filtering we will use those image rows that contain five and a half streaks resulting from six equiThese rows are situated near the bottom of the image. The distant wires. second B-scan image is an image of a human liver. It will be presented in section VII. Due to the logarithmic amplification involved, the output of the system is a logarithmically compressed version i;;(y) of the signal m(y) occuring in Eq. (1) and needed for the inverse filtering. The relationship between W(y) and m(y) is given by l,(ED Z(Y) m max

ln(D

+ ,) ,

+ 1)

- $

ln(D+l) max

I

41

(3)

.

14)

SCHOMBERG ET AL.

Fig.

3

A section of wires.

of figure

2 which

consists

of 60 rows

and shows a line

are the allowed maximal values of K(y) and m(y), resp., Here, zrnax and m and D is a parame "F er representing the 'dynamic range' of the B-scan image. In our case, D = 150. Figures 3 and 4 show the compressed and uncompressed versions of the bottom rows of the phantom, resp. III.

The Point-Spread

Function

f(y).

The basic equation (1) The already mentioned

Fig.

4

assumes intensity

a shift-invariant changes between

Uncompressed

version

42

point-spread neighbouring

of figure

3.

function lines of

LATERAL INVERSE FILTERING

a.

Fig.

5

The point-spread

Y)

function

f(y)

as defined

in

(5).

sight due to an alternate use of 3 and 4 transducer elements introduce a shift-variance which, however, can be simply removed by a scaling factor. As a point-spread function we used

; cos2(Z; ) =

f(Y)

for

lyl

5 a

(5)

/ Lo

otherwise

,

where a = 0.75 cm, see figure 5. Within the noise point-spread function describes well the observed tors. It is normalized by

s define

For later as

purposes

we note

f(y)dy

=

the

Fourier

level of the system, this images of point reflec-

.

1

(6)

transform

of f(y),

which

we

co

eiky

After

a straightforward

calculation

is well

defined

and even,

(7)

-

one finds

i-9

sin(ka) F(k) = -yyF(k)

f(y)dY

Tt2 -

see figure

(ka)*

6. We have

F(O)

=

F(E)

= F(- 9, = $

43

(8)

.

1

, .

(9)

SCHOMBERG ET AL.

Fig.

6

Fourier

transform

of the

point-spread

function.

Here, @s(k) and G,(k) are the Fourier transforms of the correlation functions of the signal and the noise, resp. In practice it is not possible to determine these exactly, and therefore one has to use approximations. We use the simplest one, which is to take Gs(k) and Q,(k) as constants. By this approximation both signal and noise are supposed to be 'white'. Then we end up with the following simplified version of the Wiener filter:

r,(y)

= 2

+.=s e

-iky

IF(

-03

The parameter p controls the F(O), i.e. on the normalization on the signal-to-noise ratio ly.

M(k)F*(k; + p*

IF(

dk

*

smoothness of rW(y). It does not depend on of f(y). The 'optimum' value of p depends and on one's taste and can be found empirical-

We implemented this method using the a length adapted to the number of columns negative values were set to zero.

Fast Fourier Transform with in the B-scan image. Possible

In the following we present a few test results obtained with the image of the phantom. The parameter p was chosen as 0.3. As criteria for the quality of the filtering procedure we use the resolution enhancement R and, somewhat more vaguely, the appearance of image degrading artifacts. As it should be, the original image was first decompressed, then filtered, and finally compressed again. The zeroes of F(k) are given by k=&n' IV.

Deconvolution

Using

the

a Fourier

'

n = 2,3,4 Transform

...

(10)

.

Technique

Deconvolution techniques based on the Fourier transform in the field of image processing [9]. The mathematical idea

is

are standard as follows.

LATERAL INVERSE FILTERING

Defining (1) the

the Fourier transform simple relation

of m(y),

M(k) Solving

for

R(k)

M(k),

= F(k)R(k)

as in

(7x one obtains

+ N(k)

and back transformation

Eq.

(11)

.

formally

from

results

in

+r(y)

= 2

e-iky

M(k)

s -co

- N(k) F(k)

(12) is not yet well suited for the However, because N(k) is unknown and F(k) has zeroes. Using

the

r,(y)

Wiener

filter

= 2

r(y)

+a, s

is

M(k)F*(k)

e -iky

-co

estimated

jF(k)l*@s(k)

ak

(12)

.

reconstruction

of r(y),

both

by

gs(k) + g,(k)

The result of this procedure applied to the image of figure 3 is shown in figure 7. A comparison of figures 3 and 7 gives a resolution enhancement of about 1.1, but we think there is no visible improvement of the image quality. The failure of the Fourier transform technique seems to be a consequence of the fact that small artifacts induced by the filtering process are strongly enhanced by the final process of logarithmic compression.

Fig.

7

Image obtained (p = 0.3).

by Wiener

filtering

of the

uncompressed

original

SCHOMBERG ET AL.

Fig.

8

Image obtained (p = 0.16).

by Wiener

filtering

Hence one might try to apply this technique images in order to avoid the critical step This procedure would be justified if there

Z(Y)

of the

original

directly to the compressed of compression after filtering. were a relation

P(y-y’)YT(y’)dy’

=

compressed

(15)

+ 3Y’)

s

connecting a logarithmically compressed signal g(y), with a logarithmically compressed reflection coefficient, Y(y). Equation (15) is analogous to Eq. (1). It holds for an object consisting of point reflectors with identical amplitude. Incidentally then, it is justified to apply the deconvolution technique directly to the compressed image of the wire phantom. The compressed point-spread function, f(y), has to be adapted to the images of the wires. We took

1 T(Y)

sin4(Ez)

for

=

(16) otherwise

0

with b = 1 cm. The result image is better than the about 1.3 to 1.4.

obtained previous

V.

the Kaczmarz

Deconvolution

]y/ s b

Using

with p = 0.16 is one; the resolution

, shown in figure 8. This enhancement is now

Method

In this section we describe and test an alternative way of estimating r(y) from (1). The idea is to approximate (1) by a discrete, linear system of equations relating a discrete version of the unknown reflection coefficient to the measured data, and then to solve this system for the unknowns. However, standard methods like Gaussian elimination are inappropriate for the latter purpose. We suggest the use of a method due to Kaczmarz [lo].

46

LATERAL INVERSE FILTERING

We start with discretizing (1). There is no unique way of doing this, but the following appears reasonable: Let y1 < y2 < . . . < yI denote the lateral coordinates of the scan lines of the underlying B-scan system. (In our case I = 123.) We have yi = y1 + (i-l) . A for some A > 0 independent of i. The scanner provides the data m(yi),l i i s I, which according to (1) are related to the unknown true reflection coefficient r(y) by

P

f(yi-y')r(y')dy'

Replacing

the

= m(Yi)

integrals

- n(Yi),

by standard

lSiS1

finite

.

(17)

sum approximations

yields

i+K A * f(yi-yj)r(yj)

c

= m(Yi)-T(Yi),

j=i-K

(18)

IrirI

.

Here the natural number K is such that f(y) is certainly zero for terms y(yi) in (18) now also incorporate the errors IYI > K * A. The noise due to replacing the integrati-ons by summations. Equations (18) involve as unknowns the terms r(yj) and n(yj). But one may hope that the latter are 'small'. We therefore decide to drop the error terms and to replace (18) by the approximate system i+K aimj

c

Uj = m(Yi)

3

15iLI

,

(19)

jzi-K

where a. .=A* 1-J

- Yj)

f(Yi

(20)

.

This is now a linear system of I equations in I+2K unknowns, ul-K ,..., UG , u1 ,..., UI , uI+l ,..., UI+K. Although not absolutely necessary, we decide to reduce the number of unknowns to the number of equations by requiring that ?-K uI+l

= . . . = u 0 = dy,) =

. . .

= u

, (21)

I+K = m(YI)

.

Of course, this assumption is somewhat arbitrary, but it is not worse than any other assumption. Equations (19) and (21) now make up a linear system of I equations in I unknowns: associated with each yi,l s i $ I, there is just one equation and one unknown. For the following discussion it is convenient to rewrite this system in matrix-vector notation as Au=b using the natural ordering of unknowns the yi. The coefficients of the matrix Aij

= aiej

,

47

,

(22)

and equations A then are

which

lSi,jSI

.

is

induced

by

(23)

SCHOMRERG ET AL.

The components (21)! 1

of u are u1 ,...,

bi

= m(yi)

-

UI , and the

aimj

c

dy,)

j
components

- c j>I

of b are

m(YI)

"i-j

together form

with (22),

3

(24) 1siSI

The system of Eqs. (19) equivalently, its matrix-vector version of (1).

(observe

.

the constraints represent the

(21) wanted

or, discrete

Depending on f(y) and the discretization procedure, the determinant of A is either zero or not. However, it is 'extremely unlikely' that it is exactly zero, and we confidently assume that it is not. Then the system (22) has a unique solution. Since the vector with components r(yi) satisfies the related system of Eqs. (l8), it is tempting to compute the exact solution of (22) and to take it:s i-th component as an approximation to r(yi). This approach, however, is doomed to failure. The reason is the 'illposed' character of the underlying convolution Eq. (1) which makes the matrix A extremely 'ill-conditioned' so that even slightest perturbations in the right hand side of (22) can, and most likely will, cause drastic changes in the solution. A detailed discussion of these phenomena is beyond the scope of this paper, and the reader is referred to e.g. [12,13]. These articles also give hints how to overcome the problem. Instead of the exact solution of (22) we should rather compute an approximate one which in addition is 'regular' in the sense that it does not oscillate too much. This sort of remedy is therefore generically referred to as regularization. There are various ways of computing an approximate (22). Here we suggest to proceed iteratively. Suppose method which, when applied to (22), produces a sequence 2, . . . converging to the exact solution of (22), for Then we might try to start with a smooth initial guess tion before the iterates begin to exhibit the unwanted exact solution. This is the basic idea. Of course we smooth initial guess available. In our case this fs U0

=

(dy,)

,...,

m(y,))

regular solution of we had an iterative of vectors u", u', any initial guess u". and stop the iteraosciliations of the should take the best

(25)

-

As an iterative method we suggest the Kaczmarz method. This method was introduced by Kaczmarz in 1937 [lo] and is thoroughly studied in [ill, where it is called projection method. It was reinvented in connection with image reconstruction from projections and given the name ART by its promoters [lh]. The Kaczmarz method solves linear systems of equations. During each iterative step, one equation of the underlying system is selected and the current iterate is updated so as to satisfy the selected equation. The One complete sweep through all equaequations are swept through cyclically. tions is called a cycle. Applied to (22) and with ul denoting the j-th component of the i-th iterate ui, the first cycle read 4: For i =

1

,...,

I

ai

=

do:

Aij

u;-'

- b.1

48

A?. =J

;

(26)

LATERAL INVERSE FILTERING

For

j = 1 ,...)

I do: ul

J

= u.

i-l

-u;A..

J

1J

;

(27)

end; end. The last iterate of the first cycle may then serve as the initial a second cycle, and so on. When the determinant of the underlying nonzero (as we are assuming), the sequence of iterates generated Kaczmarz method indeed converges to its exact solution [lo].

guess of system is by the

During the tests it was found experimentally that one cycle of the Kaczmarz method applied to (22), starting with the u0 defined in (25), did already most of the job. A few more cycles still would yield some enhancement of the resolution, which, however, would be so small that it does.not seem to justify the extra effort. If during the iteration one of the ui happened to become negative, it was immediately set to zero. In this piper applying one cycle of the Kaczmarz method to (22), starting with the u" de: fined in (25) and clipping off negative components of the iterates, will be called Kaczmarz filtering. The properties of f(y) and the discretization procedure chosen make the matrix A of (22) into a sparse, banded, symmetric Toeplitz matrix. When programming the Kaczmarz filter, one exploits of course this special structure of A. An operation count then gives about 4 KI additions and 2 KI multiplications for I >> K >> 1. To test the effectiveness of the Kaczmarz filter, we applied it to the same uncompressed original image of figure 4, in the usual row-wise manner. The compressed result is shown in figure 9. The resolution enhancement as compared with figure 3 is about 1.4 to 1.5. The artifacts in this image have appearances that differ from those of figures 7 and 9 but are not worse. Finally, we also applied the Kaczmarz filter directly to the ori-

Fig.

9

Image obtained original.

by Kaczmarz

49

filtering

of the

uncompressed

SCHOMBERG ET AL.

Fig.

IO

Image

obtained

by Kaczmarz

filtering

of the

compressed

original.

compressed image of figure 3, now using Y*(y) of (16) as a pointfunction. The resulting resolution enhancement of about 1.2 in 10 is clearly worse than that of figure 9.

ginal, spread figure VI.

Comparison

between

the

Different

Techniques

In the preceeding section we have presented some details of our special frequency and space domain techniques, together with some test results. In this section we want to compare them. We consider first the correct procedure of decompression, and compression. The simplified Wiener filter essentially fails, the Kaczmarz filter yields a resolution enhancement R of about is close to the theoretical estimate suggested by Eq. (2).

filtering, while 1.5. This

In the special case of the phantom which we used, it is also justified to filter the decompressed image directly. Equation (2) can be expected to remain essentially valid. For the image of figure 3, which has a signal-tonoise ratio of about 15, we then obtain a theoretically optimum resolution enhancement of about 1.6. In practice, the simplified Wiener filter gives R s 1.4, while the Kaczmarz filter now yields only R k 1.2. The operation counts of the various inverse filtering techniques depend on the number of image points, the width of the point-spread function comparison between the operation and other parameters chosen. A general counts of Fourier transform techniques and the Kaczmarz filtering is cum-

Fig.

11

B-scan image of a human liver. pressed version.

Fig.

12

Image

obtained

by Wiener

In our

filtering

50

notation,

of figure

it

11

is

a com-

(p = 0.16).

LATERAL INVERSE FILTERING

SCHOMBERC ET AL.

Fig.

bersome. ferences

13

Image obtained decompression

We merely note in the operation

by Kaczmarz filtering. The steps were of figure 11, deconvolution (1 cycle),

that in our case there counts.

are

A potential advantage of the Kaczmarz filtering it can also be made to work when the point-spread whereas Fourier methods fail in such a case.

no substantial lies function

as follows: compression.

dif-

in the fact that is shift-variant,

So far we have only compared the achievable resolution enhancement and operation counts. Another criterion is the amount and the appearance of artifacts that are generated by the filtering procedure. A comparison between figures 8 and 9 indicates no dramatic differences. A final decision between the different filtering techniques should be made on the basis of the filtered versions of a clinical image. This will be done in the next section. VII.

Inverse

Filtering

of an Image of a Liver

The figures presented in this section are compressed B-scan images of a human liver. Figure 11 shows the original B-scan image. It was obtained with the same technical equipment as the image of the wire phantom in figure 2. The other images are filtered versions of figure 11. Each image has been filtered in that way and with that parameter that turned out to be optimum for the wire phantom. Figure 12 shows the result of Wiener filtering. The result of Kaczmarz filtering is presented in figure 13. The fil-

52

LATERAL INVERSE FILTERING

tered images each show a slight enhancement of the resolution and different artifact patterns. We find that they contain neither more nor less information than the original image of figure 11. To rank them is a matter of taste. We like the original image most. VIII.

Concluding

Remarks

We have investigated two different ways of lateral inverse filtering and Kaczmarz filtering. Each of B-scan images, namely Wiener filtering method can be applied to compressed or decompressed images. We tried the mimicking phantom and a resulting variants on two images, showing a tissue enhance the resolution slighthuman liver, resp. The best of these variants ly but at the same time create artifact patterns of their own. In our opinion these effects of lateral inverse filtering do not justify the computational effort. The question arises whether still other approaches to lateral inverse filtering might perform better than the ones presented here. We believe they cannot. Our argument is based on two circumstances: First the noise The noise is determined by level in Eq. (1) cannot be reduced arbitrarily. One source of a modelling error measurement errors and by modelling errors. is the macroscopic inhomogeneity of tissue which causes the ultrasonic beam to propagate along curved lines instead of the assumed straight lines. This effect is not taken into account by assuming a shift-invariant point-spread Another source of a modelling error is the microscopic inhomofunction. geneity of tissue which is responsible for the appearance of speckles. Speckles are not compatible with the assumption of a 'well-behaved' reflecEvery deconvolution algorithm exploiting Eq. (1) will tion coefficient. interpret a speckle as a point reflector with the unfavourable result that the undesired specular structure is enhanced. These sources of noise are largely determined by the properties of the tissue itself. Another source of noise is due to the neglect of the third dimension outside the B-scan image. Summarizing then, the signal-to-noise ratio cannot be improved arbitrarily. The second circumstance is the logarithmic dependence of the resolution enhancement on the signal-to-noise ratio, as indicated in Eq. (2). Although (2) was derived under the assumption of a Gaussian beam shape, one can expect a similar behaviour for other, realizable beam shapes. For a realistic signal-to-noise ratio of about 15, Eq. (2) predicts a resolution enhancement of 1.6, whereas a resolution enhancement of 3, for example, would require a signal-to-noise ratio of lO,OOO! Thus, Eq. (2) expresses another fundamental limitation; in view of this limitation, the filtering methods presented here perform quite well. REFERENCES [ll

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