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Transfer functions for describing ultrasound system performance
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1979 Phys. Med. Biol. 24 146 (http://iopscience.iop.org/0031-9155/24/1/012) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 131.94.16.10 This content was downloaded on 02/10/2015 at 21:46
Please note that terms and conditions apply.
PHYS. MED. BIOL., 1979, Vol. 24, No. 1, 146-156.
Printed in Great Britain
Transfer Functions for Describing Ultrasound System Performance J. C. GORE and S. LEEMAN Department of Medical Physics, Hammersmith Hospital and Royal Postgraduate Medical School, Du Cane Road, London, W12 OHS, U.K.
Received 28 February 1978, in final form 7 August 1978 ABSTRACT.Apracticalmethod of assessing ultrasonicA-scan pulse-echo scanner performance is described, which relies upon the identification and quantitation of a n overall system transfer function. The mapping of reflecting elements into an image is then described by this transfer function, and indices of systemperformance, and particularly resolution and range accuracy, are developed in terms of the shape of the transfer function. Some advantages of this method compared to more conventional methods of assessing scanner performance are discussed.
1. Introduction
Theproliferation, in recentyears, of ultrasound pulse-echo devices for medical imaging and monitoring has necessitatedthe development of a general, quantitative method for evaluating system performance-not only in order to compare the relative merits of machines of the same type, but also, hopefully, to enable comparison between different designs (e.g. automated linear scanners against contact compound B-scanners). A useful description of performance would encourage a standardisation of equipment specifications presented by various manufacturers, andeven perhaps the laying down of production criteria and standards. It may lead to a better understanding of the pulse-echo imaging process, and possibly uncover certain features whose incorporation would result in anoverall improvement in design. And, more trivially, it would foster a better appreciation of the limits towhich scanners may be pushed so that occasionally implausible claims as to the accuracy achieved inquantitative measurement may be more realistically assessed. I n practiceanumber of systemcharacteristics may be indicated: some electronic/mechanical design features, such as the band-width and sensitivity of the amplifiers, and swept gain calibration and type, system velocity setting, echo registration,etc.,are checked andtheir accuracynoted. I n addition, acoustic transmission or transmission/reception features are usually measured, notably pulse length and beamwidth,byhydrophone or small reflector technique, and acoustic output. However, such a set of indices amounts t o little more than a system description, rather than a performance assessment. Unfortunately, theprecise elements embodied in theconcept of ‘performance’ have never been stated, but the latter is often gauged on the basis of a system’s capability for achieving spec@ tasks,under more or less ideal laboratory 0031-9155/79/0l0146+11 $01.00 @ 1979 The Institute of Physics
Transfer Functions for DescribingUltrasoundSystemPerformance
147
conditions, such as water tank scanning of simple, usually highly reflecting and specular objects. The extension of suchmeasurements to a ‘clinical’ scanning situation is generally thought to be fraught with well-nigh insurmountable difficulties and has prompted the search for tissue-equivalent ultrasound phantoms, but with limited success. However valuable testobjects may be for checking calibration and mechanical errors in the present generation of scanners, they are rather crude as monitors of more basic performance parameters. For example, there is littledoubtthatany one of anumber of testobjects would uncover registration errors in a typical conventional B-scanner, but it is by no means clear that the various ‘phantoms’ proposed for resolution measurements will recordeven the samerelativeperformance of differentscanners (Gore and Leeman 1975). Indeed, this is only one of a number of objections that may be raised against the use of phantoms in general. Since their design features are often fairly arbitrary, and not based on any underlying physical theory of the pulse-echo process, it is difficult to indicate how results from different test objectsmaybecompared, or evenrelated t o realisticscanning situations, Moreover, since there is no prescription for predicting quantitatively the result of alteration in the settings of operating controls, measurements would have of combinations of machine settings. The to be repeated for a large variety extraction of quantitative performance parameters from test-object scans is, in itself, based on arbitrary, and arguable, criteria. Finally, very little if anything a t all, may be quantitatively deduced about the change in performance to be anticipated as a result of re-design or total replacement of major system components-for example, the effect on resolution of replacing a linear by a compression amplifier cannot be predicted quantitatively from phantom tests with the unmodified system only. I n this investigation, an attempt is made to meet a t least some of the above objections. The discussion hinges on the concept that the scanning system may beregarded as a‘blackbox’transforming the interrogatedobjectinto the final image, the operation being fully specified by the transfer function of the system, tobe defined below. Particular emphasis is laid here onthe quantitative measurement of resolution, which is widely regarded as an important performance characteristic, but it is shown that other useful quantities may be For simplicity, the investigationis extractedfromthetransferfunction. restricted to A-scan systems only, but the extension to B-scanners is possible, and it is hoped to present this in a later work. The particular merit of the transfer function approach to performance assessment is that itmakes possible not only the accurateprediction of certain performance characteristics achieved in realistic scanning situations, but also enables the effect of machine control changes and re-design to be calculated. 2.
The elementalobject
I n general terms, the pulse-echo technique results in an image, or mapping, 2, since of the variations in the object’s characteristic acoustic impedance,
148
J . C. andGore
S. Leeman
reflections are generated at bothrelatively sharp and planar(i.e. surface radius of curvature 9 dominant wavelength in the acoustic pulse) interfaces between homogeneous regions of different 2, as well as from small scale fluctuations in this acoustic parameter (Gore and Leeman 1977). The fidelity of the mapping is a good indicator, therefore, of the performance of the system, and the ability to image fine structure in the impedance distribution is clearly a measure of scanner resolution. A particular approach to the quantitation of these ideas is presented here, but, for simplification, a one-dimensional model is considered, in which the impedance is assumed to be a function only of a single spatial coordinate, x, along which the ultrasound beam direction is fixed. This model is appropriate for the description of the axial resolution of A-scanning systems (Gore and Leeman 1978). A faithful ‘image’ of the impedance profile, Z(x), is defined to be any representation which is in one-to-one correspondence with it, and from which, in principle a t least, the profile may be reconstructed in unambiguousdetail. Moreover, if any physicallyreasonable Z ( x ) may be expressed as a superposition of some elemental object distribution, E(x) say, then a knowledge of the image of E ( x ) is sufficient to predict the image of a general impedance profile, provided that the mapping is to good approximation alinear process. Systemperformanceis thus seen to be embodied in the faithfulness of the image of the elemental object distribution,and it is suggested here how system performance parameters such as resolution may be derived from thisquantity.The physicalrealisation of E(x) is consequently an appropriatetestobject for the assessment of performanceparametersin practice. It is easily shown (Brown 1961, p. 63, equation (1)) thatany realistic impedance profile, Z ( x ) ,non-zero in the range (0,L ) may be written as
Z ( x )= -
J O L T
O(x’- x)dx’
where O(x) is the unit step function,
e(x):=
0, x < o 1,
x20.
It is assumed throughout that the origin of the x-axis is located at thetransducer face. Eqn (1) may be rewritten as L d lnZ(z’)] [Z(x’)O(x’-x)] dx’ Z(x) = -
jo[a
[-$lnZ(x’)] ?(x;X’) dx’. .dx ?(x;xo) is an impedance step, with magnitude of the discontinuity equal to Z(xo)(see fig. l ) , and the image of 7, normalised to unit amplitude, is the same for all values of Z ( x o )provided that the imaging process is linear. I n practice, 7,which is astep of impedance from -Z(xo) to 0 , is an air/solid planar the object 7 interface, but it maybe closely approximatedby aliquid/solidinterface. 0
TransferFunctionsforDescribingUltrasoundSystemPerformance
149
Given the normalised image of 7, the image of the general impedance profile is readily obtained from eqn (2). The distribution with unit impedance mismatch thus fulfils the requirements of the elemental test object, E , and we are led to conclude that the performance of a pulse-echo A-scanning system may
X0
X
Fig. 1. An elemental impedance step, ~ ( zx,;,).
be assessed from an analysis of the normalised display of the reflection from a plane interface. I n order to minimise any possible difficulties engendered by the presence of significant attenuation, it is proposed that the test object be a plane steel or aluminium sheet in a degassed water bath, and sufficiently thickso that a well separatedand distinctecho may be obtained from the near face. The above description is particularly useful if the image of the test object is translationally invariant; that is, it is independent of distance from the transducer. Our own experiments have shown that this is indeed so, to very good approximation, for a typical commercial scanner, with a wide range of representative transducers, over distances characteristic of those involved in diagnostic scanning situations. Implicit in the above is the assumption of linearity of the imaging process, a condition that is ensured in practice only if amplifier saturation effects are avoided and if multiple reflection artefacts are either negligible, or recognised and subtracted off. However, if some non-linear processing stage is involved, as indicated by the dependence of the (normalised) image on the value of the test object impedance jump, linearity would have to be enforced by transforming to a new output variable, or by restricting the argument to a limited range of output values over which the system remains effectively linear. It should be pointed out that translationalinvariance(stationarity)and linearity are not essential to the argument presented here, but are realistic simplifying factors. I n their absence, performance parameters, such as resolution, would depend on both distance from transducer and input/output levels of the system, a circumstance which would severely limit the utility, and cloud the interpretation, of such entities in the first place. 3.
The ideal image and the system transfer function
For a perfect scanning system, the image of the elemental object introduced above is a Dirac delta function located a t x,,,the distance of the test plane
150
J . C. Gore und S. Leernun
from the transducer face ; also in this case, the generalimpedance profile, Z ( x ) , is mappedexactly,in the finest detail,onto the display. I n practice, however, this ideal is impossible to achieve and the displayed elemental image will be a somewhat broader function, f(x- xi), non-zero only for x 3 xi. I n of reflectors. general, xi# xo,since the system may be inaccurate in its placement The performance of the scanning system is interpreted as the degree to which the displayed image of the elemental object approaches the ideal one, and a systematic procedure for quantitating the measure of the approximation is described here. The A-scan elemental image may be written generally as
where Re { } denotes the real part, and $ and g are the phase and envelope of the modulatedhighfrequencydisplay.Thex-variableisherepresumed measured from the onset of the image on the display and it is provisionally assumed that factors modifying translational invariance, such as swept-gain amplification, are not invoked at this stage. It may be noted that for most practical systems,
$(x) = ko x + $0 where g50 is a constant and ko is the magnitude of the acoustic pulse carrier wave vector. If the ideal image, 6(x- xo), is understood to be measured with respect to an origin shifted to xo, the performance of the scanning systemis then embodied in the mapping
+f
(x)-
(3)
The validity of this manipulation of the origins, with respect to which various functions are measured, will be discussed later. Formally, therefore, the system may be regarded (Gore and Leeman 1975) as a linear shift-invariant operator in image space which transforms the ideal into the displayed realisations of the object. Any analytical expression for the elemental object mapping, (3)) may be defined as the system transfer function, but we shall retainthisnomenclature for the Fourier(spatialfrequency) space representation only. The displayed image may be written as a Fourier transform :
Remembering that 6(x) = L 2n
Jrnexp (ikx)dk "00
it is apparent that, in k-space, the mapping (3) is denoted by 1+H(k). It follows immediately thatinthis representation the mappingoperationis
Transfer Functions for DescribingUltrasoundSystemPerformance
151
scalar multiplication by the transfer functionH(k),which is simply the Fourier transform of the displayed elemental object image. Thetransferfunctionis complex valued, and may be writtenin phasemodulus form : H(k) = M(k) exp(i+(k)). I n common with the usual terminology in othep fields, M(k) will be called the (MTF). It is easily shown that the reality of modulationtransferfunction f(x)implies H ( - k) = H*(k) and, consequently, that the phase function is odd and the MTF even, +( - k)
= - 4(k)
M( - k)
= M(k).
In anydescription of M(k), itis therefore sufficient to consider positive k-values only. It is convenient to adopt the normalisation
which imposes the scaling
The resolution index A more physical view of the modulation transfer function, M(k), is that it is a measure of the scanner’s ability to transmit and image the kth spatial MTF is zero, or negligibly small, frequency of the object distribution. If the beyond some spatial frequency km,,, then the A-scanner will tend to ‘smear out’ details in the object separated by distances l/kmax. Since the blurring of fine detail is intuitively associated with loss a of resolution, it is concluded that the extent of the MTF in k-space is a realistic measure of system resolving ability. There are a number of function widths which may be defined (Bracewell 1965) but a particularly useful one in the present context is the mean square width about the origin. It is proposed that an (axial) resolution index, R, be specified by 4.
R
=
[ k:/2
M2(k)dk]
’.
While the adoption of any particular width criterion t is o some extent arbitrary, it nevertheless remains true that with the choice (4),1/R represents the scale of the fine structure in Z ( x ) which the system will have relative difficulty in imaging, and consequently tend t o smooth out. Large values of R, therefore, correspond to high resolution.
J. C . Gore and X. Leeman
152
Moreover, since
ikH(k)=
J:
exp ( - ikx)dx
it readily follows that
R2 = . rom l d f / ~ 1 2 d x 0/ j m f 2 ( z ) d x SO that the resolution index may be measured in practice directly from the elemental image envelope, without recourse to Fourier transform calculations. This is a particularly strong advantage of the choice (4). The resolution index, R, may be termed an intrinsic performance parameter of the system, anddoes not necessarily correspond to theresolving performance realised intypical,practical scanningsituations (‘achieved performance’) where attenuation effects and sweptgainamplification maydestroy the translational invariance assumed here. However, the achieved resolution may be calculatedfrom the transferfunctionprovided thatthe complicating practicalfactorsareaccurately specified. Forillustrativepurposes,simple examples may be analysed, employing uncomplicated analytic approximations for the elementalimage (Gore and Leeman 1975) but, ingeneral, both the intrinsic and achieved resolutions would have to be computed by numerical techniques. I n practice,tediouscalculationsareahindrance to the useful application of the ideas presented here, and it would be preferable to employ simple approximations to the integral in ( 5 ) . The resolution index, R, is not the effective bandwidth of the pulseecho loop, but it does stand in close relationship to it, as the following argument demonstrates. The bandwidth, W, may be defined as the mean square width of the MTF, expressed relative to the mean spatial frequency, E
E It follows that
=
R
\ykM2(k) dk. =
[W2+k2]*.
A schematic representation of the relationships between R, W and k is shown in fig. 2. For a relatively long, quasi-monochromatic displayed pulse (‘narrowband’ system) W/&& 1 leading to R&E. I n this case the definition adopted for the performanceindex, R, reduces, as it should, to the usual(diffractionbased) statement that theresolution of the system is of the same order as the pulse carrier wavelength-a result which does a t least tend to confirm the internal consistency of the approach.
TransferFunctions for Describing Ultrasonic System Performance
153
It shouldbenoted thatthe transferfunctionhas been described in x-(distance) space, although a scanner actually measures time elapsed, which is transformed to distance via the machine calibration. Such a description is preferable since it accords with the conventional view that resolutionisa
k
k
Fig. 2. A typical MTF, M ( k ) ,of an ultrasonic scanner showing the relationship between W , the bandwidth, R , the resolution, and h, the mean spatial frequency.
distanceconcept, but foraccuratemeasurements of resolution the system velocity should f i s t be set to the velocity of sound in the measuring medium, although it is simple to calculate a scale correction when the system velocity is inappropriate but known. Furthermore, the displayed pulse is apparently shortened by a factor of two, which corresponds to the well known fact that echoes from adjacent reflectors return t o the transducer witha time separation which corresponds to twice the distance between the reflectors. The resolution index described above provides an appropriate description of the ability of the system t o resolve true distances inthe object.An alternative formulation of the systemtransferfunction could be developed interms of itstime domain response (corresponding t o the temporal frequency response of the pulse-echo loop) but such a description would necessarily relate to an object apparently scaled in time bya factor of two (as implied, for example, in Gore and Leeman 1977, equation (10)). The range accuracy The transfer function contains all information on the imaging performance of the system (subject to the constraints mentioned above) and, in principle, not only are all intrinsicperformance indices derivable from it, but also achieved characteristics are calculable, provided sufficient additionalinformationis supplied. The intrinsic resolution of the system was seen above to be determined by the modulus of the transfer function, and the natural question arises as to what performance characteristic is determined by its phase. It is shown here that the intrinsic range accuracy of the system, viz. the ultimate precision with which it can locate the absolute distance of an interface in a homogeneous by the dependence of the phase medium of known velocity, is determined function on the spatial frequency. 5.
154
J . C . Gore and X. Leeman
It has been indicated that the displayed elemental image operation of the transfer function on the ideal image, f(x-Xi) = 2T
is given by the
I"
M(k) exp (i$(k)) exp [ik(x -xo)] dk.
"m
It is now explicitly underlined that the displayed image may not be located a t t h e samerangeas the idealimage.Causality will demand that xi >xo. Moreover, the system transfer function is temporarily regarded here as being exactly known and not calculated by the procedureoutlinedinaprevious section. The integral in eqn (6) extends formally over the entire k-space, but the MTF is expected on physical grounds to become negligibly small for large k, and vanishfor some I k I > k,,, so that theentire contribution to theexpression is from some finite k-range only. If $ ( k ) varies smoothly and not too rapidly throughout this region, and remembering that it must be an odd function, then it may be expanded in a Taylor expansion about k = 0 in the k-range of interest : $(k) = k(d$/dk) Ik=O k3(d3$/dk3)1k-O ....
+
+
The displayed image is then given by
S ( k )exp [ik(x- x.
+ $0')]
dk = h(x - x.
where F ( k ) is some complex functionwithmodulusequal where $0' E d$/dk lk=O.
to the
MTF,
and
A comparison of eqns (6) and ( 7 ) shows that the elemental displayed image is shifted with respect to the ideal by an amount ( b o t , and range accuracy is thus seen to be determined by the slope of the phase function at the origin. As pointed out above, causality demands that x. - xi be non-positive, which in turn constrains the slope of the phase function to be negative, or a t most zero, through the origin. The rather heuristic arguments presented here also make it clear that the previously mentioned apparent disregard for the precise value of the origin of the displayed elemental image has the consequence that the phase function, as measured by the method indicated above, possibly deviates from the true one byanamountequalto However, the measuredtransferfunction, which corresponds to F ( k ) , provides an accurate MTF, so that the resolution index, as defined, is not influenced. The true phase function may be derived from the empirical one, provided that thevalue of $o' is fixed by accuratemeasurements of xi and xo. I n practice, such measurements will not be considered necessary in routine system assessments as range accuracy, as such, is of minor importance in medical pulserelative distancemeasureecho systems, where the emphasisisonaccurate ments. An exception to this is the case for which stationarity does not hold,
Transfer Functions for DescribingUltrasound System Performance
155
and where the range accuracy, as defined here, varies with distance from the transducer-a situation which manifests itself as distortion of the displayed image.
6. Conclusions
Thesystem performance of an ultrasonic pulse-echo A-scan has been investigated from a fundamental point of view, and it has been shown how suitable indices of performance, and in particularof resolution, may be specified from measurements taken with a simple test object. The method described has significant practical and analytical advantages over more conventional approaches involving test objects, and in particular it provides considerable insight into quantifying the effects of changes in the scanner design or conditions of operation. It further provides an understanding of the relative influences of different segments of the system since the overall transfer function canusually be considered as a cascade of component elemental responses, each of which in principle can be measured separately. For example, two important determinants of the total system transfer properties are the impulse responses of the transducer in transmission and reception, which can each be measured or analysedtheoretically (Redwood 1961). It is further believed that the relatively formal approach to assessing scanner performance outlined here produces a resolution index which conforms closely with subjective andintuitive concepts of image quality proposed elsewhere. The extension of the ideasintroduced here to more complex imaging,suchas B-scans, and particular examples of their application, willfollow in a later publication.
RESUME Fonctions de transfert pour dQcrire la performance des systbmes d’ultra-sons Description d’une methode pratique pour Qvaluer la performance d’un detecteur iL echo d’impulsions A-balayage ultrasonique, qui se base sur l’identification et la quantification d’une fonction de transfert d’un syst&me global. La correspondance d’B1Qmentsr6flQchissants en une image est alors decrite par cette fonction de transfert; les indices de performance du systirme et particulibrement la precision de la gamme et de la resolution sont dQveloppQs en termes de la forme de la fonction de transfert. Examen de quelques avantages de cette mQthodeen comparaison avec des mQthodes plus conventionnelles de determination de la performance du dQtecteur.
ZUSAMMENFASSUNG ubertragungsfunktionen fur die Beschreibung der LeistungsfahigkeitvonUltraschallsystemen Eine praktische Methode zur Bewertung der Leistungsfahigkeit eines Eltraschallabtasters mit A-Abtast-Impulsechowird beschrieben. Diese beruht auf der Identifizierung und Quantifizierung einer Ubertragungsfunktion fur das gesamte System. Die Aufzeichnung reflektierender Elemente in einem Bild wird dann durch diese Ubertragungsfunktiondargestellt.Kennwerte fur die Leistungsfiihigkeit des Systems, insbesondere Auflosung und Bereichgenauigkeit, werden von dem Verlauf der Ubertragungsfunktionabgeleitet. Einige Vorteile dieser Methode verglichen mit bisher ublichen Methoden zur Bewertung der Leistungsfiihigkeit von Abtastern werden diskutiert.
156
TransferFunctionsforDescribingUltrasoundSystemPerformance
REFEREKCES BRACEWELL, R . , 1965, The Fourier Transform and its Applications (Xew York: McGrawHill). BROWN, B. M., 1961, The Mathematical Theory of Linear Systems (London: Chapman and Hall). GORE,J. C., and LEEMAN, S.,1975, in Proc. 2nd European Congr. on Ultrasonicsin Xedicine Ed. de Vlieger et al. (Amsterdam: Exoerpta Medica) pp. 197-206. GORE, J. C., and LEEMAN, S.,1977, Phys. Med. Biol., 22, 317. GORE, J. C., andLEEMAN, S., 1978, in TheEcaluation and Calibration of Ultrasonic Transducers, Ed. M. Silk (Guildford: IPC). REDWOOD, M., 1961, J . Acoust. Soc. Am., 33, 527.
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My IOPscience
Transfer functions for describing ultrasound system performance
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1979 Phys. Med. Biol. 24 146 (http://iopscience.iop.org/0031-9155/24/1/012) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 131.94.16.10 This content was downloaded on 02/10/2015 at 21:46
Please note that terms and conditions apply.
PHYS. MED. BIOL., 1979, Vol. 24, No. 1, 146-156.
Printed in Great Britain
Transfer Functions for Describing Ultrasound System Performance J. C. GORE and S. LEEMAN Department of Medical Physics, Hammersmith Hospital and Royal Postgraduate Medical School, Du Cane Road, London, W12 OHS, U.K.
Received 28 February 1978, in final form 7 August 1978 ABSTRACT.Apracticalmethod of assessing ultrasonicA-scan pulse-echo scanner performance is described, which relies upon the identification and quantitation of a n overall system transfer function. The mapping of reflecting elements into an image is then described by this transfer function, and indices of systemperformance, and particularly resolution and range accuracy, are developed in terms of the shape of the transfer function. Some advantages of this method compared to more conventional methods of assessing scanner performance are discussed.
1. Introduction
Theproliferation, in recentyears, of ultrasound pulse-echo devices for medical imaging and monitoring has necessitatedthe development of a general, quantitative method for evaluating system performance-not only in order to compare the relative merits of machines of the same type, but also, hopefully, to enable comparison between different designs (e.g. automated linear scanners against contact compound B-scanners). A useful description of performance would encourage a standardisation of equipment specifications presented by various manufacturers, andeven perhaps the laying down of production criteria and standards. It may lead to a better understanding of the pulse-echo imaging process, and possibly uncover certain features whose incorporation would result in anoverall improvement in design. And, more trivially, it would foster a better appreciation of the limits towhich scanners may be pushed so that occasionally implausible claims as to the accuracy achieved inquantitative measurement may be more realistically assessed. I n practiceanumber of systemcharacteristics may be indicated: some electronic/mechanical design features, such as the band-width and sensitivity of the amplifiers, and swept gain calibration and type, system velocity setting, echo registration,etc.,are checked andtheir accuracynoted. I n addition, acoustic transmission or transmission/reception features are usually measured, notably pulse length and beamwidth,byhydrophone or small reflector technique, and acoustic output. However, such a set of indices amounts t o little more than a system description, rather than a performance assessment. Unfortunately, theprecise elements embodied in theconcept of ‘performance’ have never been stated, but the latter is often gauged on the basis of a system’s capability for achieving spec@ tasks,under more or less ideal laboratory 0031-9155/79/0l0146+11 $01.00 @ 1979 The Institute of Physics
Transfer Functions for DescribingUltrasoundSystemPerformance
147
conditions, such as water tank scanning of simple, usually highly reflecting and specular objects. The extension of suchmeasurements to a ‘clinical’ scanning situation is generally thought to be fraught with well-nigh insurmountable difficulties and has prompted the search for tissue-equivalent ultrasound phantoms, but with limited success. However valuable testobjects may be for checking calibration and mechanical errors in the present generation of scanners, they are rather crude as monitors of more basic performance parameters. For example, there is littledoubtthatany one of anumber of testobjects would uncover registration errors in a typical conventional B-scanner, but it is by no means clear that the various ‘phantoms’ proposed for resolution measurements will recordeven the samerelativeperformance of differentscanners (Gore and Leeman 1975). Indeed, this is only one of a number of objections that may be raised against the use of phantoms in general. Since their design features are often fairly arbitrary, and not based on any underlying physical theory of the pulse-echo process, it is difficult to indicate how results from different test objectsmaybecompared, or evenrelated t o realisticscanning situations, Moreover, since there is no prescription for predicting quantitatively the result of alteration in the settings of operating controls, measurements would have of combinations of machine settings. The to be repeated for a large variety extraction of quantitative performance parameters from test-object scans is, in itself, based on arbitrary, and arguable, criteria. Finally, very little if anything a t all, may be quantitatively deduced about the change in performance to be anticipated as a result of re-design or total replacement of major system components-for example, the effect on resolution of replacing a linear by a compression amplifier cannot be predicted quantitatively from phantom tests with the unmodified system only. I n this investigation, an attempt is made to meet a t least some of the above objections. The discussion hinges on the concept that the scanning system may beregarded as a‘blackbox’transforming the interrogatedobjectinto the final image, the operation being fully specified by the transfer function of the system, tobe defined below. Particular emphasis is laid here onthe quantitative measurement of resolution, which is widely regarded as an important performance characteristic, but it is shown that other useful quantities may be For simplicity, the investigationis extractedfromthetransferfunction. restricted to A-scan systems only, but the extension to B-scanners is possible, and it is hoped to present this in a later work. The particular merit of the transfer function approach to performance assessment is that itmakes possible not only the accurateprediction of certain performance characteristics achieved in realistic scanning situations, but also enables the effect of machine control changes and re-design to be calculated. 2.
The elementalobject
I n general terms, the pulse-echo technique results in an image, or mapping, 2, since of the variations in the object’s characteristic acoustic impedance,
148
J . C. andGore
S. Leeman
reflections are generated at bothrelatively sharp and planar(i.e. surface radius of curvature 9 dominant wavelength in the acoustic pulse) interfaces between homogeneous regions of different 2, as well as from small scale fluctuations in this acoustic parameter (Gore and Leeman 1977). The fidelity of the mapping is a good indicator, therefore, of the performance of the system, and the ability to image fine structure in the impedance distribution is clearly a measure of scanner resolution. A particular approach to the quantitation of these ideas is presented here, but, for simplification, a one-dimensional model is considered, in which the impedance is assumed to be a function only of a single spatial coordinate, x, along which the ultrasound beam direction is fixed. This model is appropriate for the description of the axial resolution of A-scanning systems (Gore and Leeman 1978). A faithful ‘image’ of the impedance profile, Z(x), is defined to be any representation which is in one-to-one correspondence with it, and from which, in principle a t least, the profile may be reconstructed in unambiguousdetail. Moreover, if any physicallyreasonable Z ( x ) may be expressed as a superposition of some elemental object distribution, E(x) say, then a knowledge of the image of E ( x ) is sufficient to predict the image of a general impedance profile, provided that the mapping is to good approximation alinear process. Systemperformanceis thus seen to be embodied in the faithfulness of the image of the elemental object distribution,and it is suggested here how system performance parameters such as resolution may be derived from thisquantity.The physicalrealisation of E(x) is consequently an appropriatetestobject for the assessment of performanceparametersin practice. It is easily shown (Brown 1961, p. 63, equation (1)) thatany realistic impedance profile, Z ( x ) ,non-zero in the range (0,L ) may be written as
Z ( x )= -
J O L T
O(x’- x)dx’
where O(x) is the unit step function,
e(x):=
0, x < o 1,
x20.
It is assumed throughout that the origin of the x-axis is located at thetransducer face. Eqn (1) may be rewritten as L d lnZ(z’)] [Z(x’)O(x’-x)] dx’ Z(x) = -
jo[a
[-$lnZ(x’)] ?(x;X’) dx’. .dx ?(x;xo) is an impedance step, with magnitude of the discontinuity equal to Z(xo)(see fig. l ) , and the image of 7, normalised to unit amplitude, is the same for all values of Z ( x o )provided that the imaging process is linear. I n practice, 7,which is astep of impedance from -Z(xo) to 0 , is an air/solid planar the object 7 interface, but it maybe closely approximatedby aliquid/solidinterface. 0
TransferFunctionsforDescribingUltrasoundSystemPerformance
149
Given the normalised image of 7, the image of the general impedance profile is readily obtained from eqn (2). The distribution with unit impedance mismatch thus fulfils the requirements of the elemental test object, E , and we are led to conclude that the performance of a pulse-echo A-scanning system may
X0
X
Fig. 1. An elemental impedance step, ~ ( zx,;,).
be assessed from an analysis of the normalised display of the reflection from a plane interface. I n order to minimise any possible difficulties engendered by the presence of significant attenuation, it is proposed that the test object be a plane steel or aluminium sheet in a degassed water bath, and sufficiently thickso that a well separatedand distinctecho may be obtained from the near face. The above description is particularly useful if the image of the test object is translationally invariant; that is, it is independent of distance from the transducer. Our own experiments have shown that this is indeed so, to very good approximation, for a typical commercial scanner, with a wide range of representative transducers, over distances characteristic of those involved in diagnostic scanning situations. Implicit in the above is the assumption of linearity of the imaging process, a condition that is ensured in practice only if amplifier saturation effects are avoided and if multiple reflection artefacts are either negligible, or recognised and subtracted off. However, if some non-linear processing stage is involved, as indicated by the dependence of the (normalised) image on the value of the test object impedance jump, linearity would have to be enforced by transforming to a new output variable, or by restricting the argument to a limited range of output values over which the system remains effectively linear. It should be pointed out that translationalinvariance(stationarity)and linearity are not essential to the argument presented here, but are realistic simplifying factors. I n their absence, performance parameters, such as resolution, would depend on both distance from transducer and input/output levels of the system, a circumstance which would severely limit the utility, and cloud the interpretation, of such entities in the first place. 3.
The ideal image and the system transfer function
For a perfect scanning system, the image of the elemental object introduced above is a Dirac delta function located a t x,,,the distance of the test plane
150
J . C. Gore und S. Leernun
from the transducer face ; also in this case, the generalimpedance profile, Z ( x ) , is mappedexactly,in the finest detail,onto the display. I n practice, however, this ideal is impossible to achieve and the displayed elemental image will be a somewhat broader function, f(x- xi), non-zero only for x 3 xi. I n of reflectors. general, xi# xo,since the system may be inaccurate in its placement The performance of the scanning system is interpreted as the degree to which the displayed image of the elemental object approaches the ideal one, and a systematic procedure for quantitating the measure of the approximation is described here. The A-scan elemental image may be written generally as
where Re { } denotes the real part, and $ and g are the phase and envelope of the modulatedhighfrequencydisplay.Thex-variableisherepresumed measured from the onset of the image on the display and it is provisionally assumed that factors modifying translational invariance, such as swept-gain amplification, are not invoked at this stage. It may be noted that for most practical systems,
$(x) = ko x + $0 where g50 is a constant and ko is the magnitude of the acoustic pulse carrier wave vector. If the ideal image, 6(x- xo), is understood to be measured with respect to an origin shifted to xo, the performance of the scanning systemis then embodied in the mapping
+f
(x)-
(3)
The validity of this manipulation of the origins, with respect to which various functions are measured, will be discussed later. Formally, therefore, the system may be regarded (Gore and Leeman 1975) as a linear shift-invariant operator in image space which transforms the ideal into the displayed realisations of the object. Any analytical expression for the elemental object mapping, (3)) may be defined as the system transfer function, but we shall retainthisnomenclature for the Fourier(spatialfrequency) space representation only. The displayed image may be written as a Fourier transform :
Remembering that 6(x) = L 2n
Jrnexp (ikx)dk "00
it is apparent that, in k-space, the mapping (3) is denoted by 1+H(k). It follows immediately thatinthis representation the mappingoperationis
Transfer Functions for DescribingUltrasoundSystemPerformance
151
scalar multiplication by the transfer functionH(k),which is simply the Fourier transform of the displayed elemental object image. Thetransferfunctionis complex valued, and may be writtenin phasemodulus form : H(k) = M(k) exp(i+(k)). I n common with the usual terminology in othep fields, M(k) will be called the (MTF). It is easily shown that the reality of modulationtransferfunction f(x)implies H ( - k) = H*(k) and, consequently, that the phase function is odd and the MTF even, +( - k)
= - 4(k)
M( - k)
= M(k).
In anydescription of M(k), itis therefore sufficient to consider positive k-values only. It is convenient to adopt the normalisation
which imposes the scaling
The resolution index A more physical view of the modulation transfer function, M(k), is that it is a measure of the scanner’s ability to transmit and image the kth spatial MTF is zero, or negligibly small, frequency of the object distribution. If the beyond some spatial frequency km,,, then the A-scanner will tend to ‘smear out’ details in the object separated by distances l/kmax. Since the blurring of fine detail is intuitively associated with loss a of resolution, it is concluded that the extent of the MTF in k-space is a realistic measure of system resolving ability. There are a number of function widths which may be defined (Bracewell 1965) but a particularly useful one in the present context is the mean square width about the origin. It is proposed that an (axial) resolution index, R, be specified by 4.
R
=
[ k:/2
M2(k)dk]
’.
While the adoption of any particular width criterion t is o some extent arbitrary, it nevertheless remains true that with the choice (4),1/R represents the scale of the fine structure in Z ( x ) which the system will have relative difficulty in imaging, and consequently tend t o smooth out. Large values of R, therefore, correspond to high resolution.
J. C . Gore and X. Leeman
152
Moreover, since
ikH(k)=
J:
exp ( - ikx)dx
it readily follows that
R2 = . rom l d f / ~ 1 2 d x 0/ j m f 2 ( z ) d x SO that the resolution index may be measured in practice directly from the elemental image envelope, without recourse to Fourier transform calculations. This is a particularly strong advantage of the choice (4). The resolution index, R, may be termed an intrinsic performance parameter of the system, anddoes not necessarily correspond to theresolving performance realised intypical,practical scanningsituations (‘achieved performance’) where attenuation effects and sweptgainamplification maydestroy the translational invariance assumed here. However, the achieved resolution may be calculatedfrom the transferfunctionprovided thatthe complicating practicalfactorsareaccurately specified. Forillustrativepurposes,simple examples may be analysed, employing uncomplicated analytic approximations for the elementalimage (Gore and Leeman 1975) but, ingeneral, both the intrinsic and achieved resolutions would have to be computed by numerical techniques. I n practice,tediouscalculationsareahindrance to the useful application of the ideas presented here, and it would be preferable to employ simple approximations to the integral in ( 5 ) . The resolution index, R, is not the effective bandwidth of the pulseecho loop, but it does stand in close relationship to it, as the following argument demonstrates. The bandwidth, W, may be defined as the mean square width of the MTF, expressed relative to the mean spatial frequency, E
E It follows that
=
R
\ykM2(k) dk. =
[W2+k2]*.
A schematic representation of the relationships between R, W and k is shown in fig. 2. For a relatively long, quasi-monochromatic displayed pulse (‘narrowband’ system) W/&& 1 leading to R&E. I n this case the definition adopted for the performanceindex, R, reduces, as it should, to the usual(diffractionbased) statement that theresolution of the system is of the same order as the pulse carrier wavelength-a result which does a t least tend to confirm the internal consistency of the approach.
TransferFunctions for Describing Ultrasonic System Performance
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It shouldbenoted thatthe transferfunctionhas been described in x-(distance) space, although a scanner actually measures time elapsed, which is transformed to distance via the machine calibration. Such a description is preferable since it accords with the conventional view that resolutionisa
k
k
Fig. 2. A typical MTF, M ( k ) ,of an ultrasonic scanner showing the relationship between W , the bandwidth, R , the resolution, and h, the mean spatial frequency.
distanceconcept, but foraccuratemeasurements of resolution the system velocity should f i s t be set to the velocity of sound in the measuring medium, although it is simple to calculate a scale correction when the system velocity is inappropriate but known. Furthermore, the displayed pulse is apparently shortened by a factor of two, which corresponds to the well known fact that echoes from adjacent reflectors return t o the transducer witha time separation which corresponds to twice the distance between the reflectors. The resolution index described above provides an appropriate description of the ability of the system t o resolve true distances inthe object.An alternative formulation of the systemtransferfunction could be developed interms of itstime domain response (corresponding t o the temporal frequency response of the pulse-echo loop) but such a description would necessarily relate to an object apparently scaled in time bya factor of two (as implied, for example, in Gore and Leeman 1977, equation (10)). The range accuracy The transfer function contains all information on the imaging performance of the system (subject to the constraints mentioned above) and, in principle, not only are all intrinsicperformance indices derivable from it, but also achieved characteristics are calculable, provided sufficient additionalinformationis supplied. The intrinsic resolution of the system was seen above to be determined by the modulus of the transfer function, and the natural question arises as to what performance characteristic is determined by its phase. It is shown here that the intrinsic range accuracy of the system, viz. the ultimate precision with which it can locate the absolute distance of an interface in a homogeneous by the dependence of the phase medium of known velocity, is determined function on the spatial frequency. 5.
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J . C . Gore and X. Leeman
It has been indicated that the displayed elemental image operation of the transfer function on the ideal image, f(x-Xi) = 2T
is given by the
I"
M(k) exp (i$(k)) exp [ik(x -xo)] dk.
"m
It is now explicitly underlined that the displayed image may not be located a t t h e samerangeas the idealimage.Causality will demand that xi >xo. Moreover, the system transfer function is temporarily regarded here as being exactly known and not calculated by the procedureoutlinedinaprevious section. The integral in eqn (6) extends formally over the entire k-space, but the MTF is expected on physical grounds to become negligibly small for large k, and vanishfor some I k I > k,,, so that theentire contribution to theexpression is from some finite k-range only. If $ ( k ) varies smoothly and not too rapidly throughout this region, and remembering that it must be an odd function, then it may be expanded in a Taylor expansion about k = 0 in the k-range of interest : $(k) = k(d$/dk) Ik=O k3(d3$/dk3)1k-O ....
+
+
The displayed image is then given by
S ( k )exp [ik(x- x.
+ $0')]
dk = h(x - x.
where F ( k ) is some complex functionwithmodulusequal where $0' E d$/dk lk=O.
to the
MTF,
and
A comparison of eqns (6) and ( 7 ) shows that the elemental displayed image is shifted with respect to the ideal by an amount ( b o t , and range accuracy is thus seen to be determined by the slope of the phase function at the origin. As pointed out above, causality demands that x. - xi be non-positive, which in turn constrains the slope of the phase function to be negative, or a t most zero, through the origin. The rather heuristic arguments presented here also make it clear that the previously mentioned apparent disregard for the precise value of the origin of the displayed elemental image has the consequence that the phase function, as measured by the method indicated above, possibly deviates from the true one byanamountequalto However, the measuredtransferfunction, which corresponds to F ( k ) , provides an accurate MTF, so that the resolution index, as defined, is not influenced. The true phase function may be derived from the empirical one, provided that thevalue of $o' is fixed by accuratemeasurements of xi and xo. I n practice, such measurements will not be considered necessary in routine system assessments as range accuracy, as such, is of minor importance in medical pulserelative distancemeasureecho systems, where the emphasisisonaccurate ments. An exception to this is the case for which stationarity does not hold,
Transfer Functions for DescribingUltrasound System Performance
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and where the range accuracy, as defined here, varies with distance from the transducer-a situation which manifests itself as distortion of the displayed image.
6. Conclusions
Thesystem performance of an ultrasonic pulse-echo A-scan has been investigated from a fundamental point of view, and it has been shown how suitable indices of performance, and in particularof resolution, may be specified from measurements taken with a simple test object. The method described has significant practical and analytical advantages over more conventional approaches involving test objects, and in particular it provides considerable insight into quantifying the effects of changes in the scanner design or conditions of operation. It further provides an understanding of the relative influences of different segments of the system since the overall transfer function canusually be considered as a cascade of component elemental responses, each of which in principle can be measured separately. For example, two important determinants of the total system transfer properties are the impulse responses of the transducer in transmission and reception, which can each be measured or analysedtheoretically (Redwood 1961). It is further believed that the relatively formal approach to assessing scanner performance outlined here produces a resolution index which conforms closely with subjective andintuitive concepts of image quality proposed elsewhere. The extension of the ideasintroduced here to more complex imaging,suchas B-scans, and particular examples of their application, willfollow in a later publication.
RESUME Fonctions de transfert pour dQcrire la performance des systbmes d’ultra-sons Description d’une methode pratique pour Qvaluer la performance d’un detecteur iL echo d’impulsions A-balayage ultrasonique, qui se base sur l’identification et la quantification d’une fonction de transfert d’un syst&me global. La correspondance d’B1Qmentsr6flQchissants en une image est alors decrite par cette fonction de transfert; les indices de performance du systirme et particulibrement la precision de la gamme et de la resolution sont dQveloppQs en termes de la forme de la fonction de transfert. Examen de quelques avantages de cette mQthodeen comparaison avec des mQthodes plus conventionnelles de determination de la performance du dQtecteur.
ZUSAMMENFASSUNG ubertragungsfunktionen fur die Beschreibung der LeistungsfahigkeitvonUltraschallsystemen Eine praktische Methode zur Bewertung der Leistungsfahigkeit eines Eltraschallabtasters mit A-Abtast-Impulsechowird beschrieben. Diese beruht auf der Identifizierung und Quantifizierung einer Ubertragungsfunktion fur das gesamte System. Die Aufzeichnung reflektierender Elemente in einem Bild wird dann durch diese Ubertragungsfunktiondargestellt.Kennwerte fur die Leistungsfiihigkeit des Systems, insbesondere Auflosung und Bereichgenauigkeit, werden von dem Verlauf der Ubertragungsfunktionabgeleitet. Einige Vorteile dieser Methode verglichen mit bisher ublichen Methoden zur Bewertung der Leistungsfiihigkeit von Abtastern werden diskutiert.
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REFEREKCES BRACEWELL, R . , 1965, The Fourier Transform and its Applications (Xew York: McGrawHill). BROWN, B. M., 1961, The Mathematical Theory of Linear Systems (London: Chapman and Hall). GORE,J. C., and LEEMAN, S.,1975, in Proc. 2nd European Congr. on Ultrasonicsin Xedicine Ed. de Vlieger et al. (Amsterdam: Exoerpta Medica) pp. 197-206. GORE, J. C., and LEEMAN, S.,1977, Phys. Med. Biol., 22, 317. GORE, J. C., andLEEMAN, S., 1978, in TheEcaluation and Calibration of Ultrasonic Transducers, Ed. M. Silk (Guildford: IPC). REDWOOD, M., 1961, J . Acoust. Soc. Am., 33, 527.
