Generalized NEQ for assessment of ultrasound image quality

Transcription

1 Generalized NEQ for assessent of ultrasound iage quality Roger J. Zep *, Craig K. Aey, and Michael F. Insana Departent of Bioedical Engineering, University of California, Davis, 9566 ABSTRACT An inforation-theoretic fraework for assessing and predicting ultrasound syste perforance for detection tasks is outlined. Current odels of iage quality for ultrasound detection tasks ake soe stringent assuptions, including large target area, that place liits on the applicaility of the theory. New odels of iage quality for ultrasound systes are proposed ased on the ideal oserver that account for noise, syste, and oject properties. One result is an expression for the ideal oserver detectaility that is a generalization of Noise-Equivalent Quanta (NEQ), a easure used y photon iaging odalities. The detection signal-to-noise ratio is shown to e an integration of the generalized NEQ weighted y the spectral variance of the target signal (in contrast to the squared agnitude of the Fourier transfor of the signal, as is the case with other odalities). This reflects that ultrasound systes are sensitive not to the agnitude of the ediu paraeters ut rather the variance (spatial fluctuations) of these quantities. The resulting fraework is aenale to easureent and prediction of syste perforance. The theory is used to predict the inforation content of the ultrasonic ea at various field points. New strategies are revealed for processing RF data that could iprove detection of lesions. Keywords: Ideal oserver, noise-equivalent quanta, lesion, detection. INTRODUCTION AND MOTIVATION At the frontier of cancer iaging is the identification and characterization of non-palpale neoplass. Ultrasound systes are playing a ore proinent role in the detection and anageent of cancer, as well as in understanding the asic echaniss of disease progression. Central to effective iaging syste design is the aility to rigorously and quantitatively characterize syste perforance. The concept of iage quality can e ade precise when associated with the aility of iaging systes to perfor soe task 3. We focus our attention on cancer applications, where the task is to say whether a neoplas is present or asent, given an iage or set of iages. The approach taken this paper is to provide a way to optiize ultrasound systes specifically for lesion detection tasks. In other words, we can tailor the syste to the task to optiize relevant inforation content in the raw data. We do this y outlining a foralis to quantitatively characterize ultrasound syste perforance ased on odels of the syste, the noise, as well as accounting for statistical variaility in the ody. We use an inforation-theoretic approach ased on the ideal oserver 4 for ultrasound detection tasks. To review the concept of an ideal oserver, we first define what is eant y an oserver. For detection tasks, we define an oserver as an individual or entity that akes a inary decision given a set of iage data represented y the vector g. An oserver ay often e odeled y a function λ(g) of the iage data. Given an iage, the oserver synthesizes a scalar test statistic λ(g), then copares this nuer with a threshold to ake a decision. Over an ensele of iages where a signal is present, λ will e statistically distriuted aout soe ean. Likewise, there will e a corresponding distriution when the signal is asent. Different oservers will give rise to different distriutions of λ. Upon thresholding, ost oservers will ake decision errors. The ost widespread way of characterizing oserver perforance for inary detection tasks is to plot the proaility of a true positive for a given false positive proaility in a Receiver Operating Characteristic (ROC) curve. An ideal oserver has the optiu ROC curve of all oservers, that is, it has the axiu possile true positive proaility for every false positive level. Classical statistical decision theory has shown that the optial oserver test statistic is the likelihood ratio, or a onotonic transforation of it, such as the log-likelihood ratio 5. Oservers for signal * E-ail: rjzep@ucdavis.edu; phone: (530)

2 detection tasks, and indeed any task in edical iaging task can e influenced y a nuer of factors. First, there is oject variaility or variations in the ody that can oscure diagnostic interpretation. Second, syste lur and noise, as well as confounding artifacts liit the type of inforation that can e collected. Third, the raw signal collected y the iaging syste will e processed. This processing ay incur soe loss of inforation. Fourth, the processed data will e displayed in soe edia forat, and fifth, the eye-rain syste of the individual reading the iages will influence interpretation. The ideal oserver is not concerned with factors 4 and 5, ut rather is concerned with how oject variaility, syste design, and processing algoriths affect the detection task. Understanding the ideal oserver is iportant for syste and algorith design. The goal is to ake ultrasound systes etter than they now are at detecting cancerous lesions. Optiizing a syste for this task can e viewed as a two-step procedure. First, one should design the syste so as to acquire as uch task-relevant inforation availale as possile in the raw signal. Second, one should design processing algoriths and display the processed data in a way that akes inforation aout the task optially availale to huan oservers. In other words, we want the huan oserver to perfor as nearly as possile to the ideal oserver upper ound. Although the ultiate goal is to iprove perforance of huan oservers y iproving design and algoriths, this paper does not attept to odel huan oserver perforance. We are ainly concerned with quantifying the upper ound on inforation content. Future work will attept to integrate oth huan and ideal perforance to optiize for the task. In soe cases it ay e necessary to sacrifice ideal oserver perforance to iprove huan oserver perforance. We should ention that true clinical tasks have any coplexities, and that we only concern ourselves with odeling essential features of true clinical tasks. Besides quantifying inforation content, the ideal oserver is also iportant for another reason. The ideal oserver indicates strategies for how to use raw data to ake a decision that could help huan oservers. An iage processing algorith that could iic the ideal oserver s strategy could e very valuale. Currently, B-ode ultrasound systes use envelope detection, a process that discards phase inforation. Does this procedure incur a loss of inforation relevant to the detection task? Are there any other practical ways of processing the data? These questions need a fundaental fraework to e answered. We egin our discussion with a review of current odels of iage quality in the literature. We show the need for a new odel, and develop a fraework applicale to ore realistic situations. We then draw connections to current literature regarding Generalized Noise-Equivalent Quanta (GNEQ), a ter first coined y Barrett et al. 6 as a generalization of Noise-Equivalent Quanta (NEQ) first introduced y Shaw 7 for photon iaging systes. The interpretation for Generalized NEQ for ultrasound systes is given, and an explanation is given regarding how perforance etrics for ultrasound systes differ fro photon iaging odalities. We show that GNEQ characterizes the ultrasound syste for a given tissue type. Our results provide a practical way to not only predict, ut to easure the effectiveness of design or eaforing strategies in various scattering edia.. LITERATURE REVIEW Soe iportant contriutions to the area of ultrasonic iage quality have een ade in past literature. Seinal papers y Wagner and Sith and colleagues 8,9,0 have outlined fundaental odels for signal statistics, and a siple odel for the ideal oserver given specialized circustances. Here we give a didactic overview of the salient features of these findings, clarify assuptions ade, and otivate the need for new odels. The Sith and Wagner approach egan with an assuption that echo signals could e odeled as a rando walk in the coplex plane. For large nuers of rando scatterers per pulse volue, envelope-detected iage data v (representing N pixels) was shown to exhiit Rayleigh statistics, characterized y a single paraeter, ψ: pdf v = e ψ N v / ψ v ψ ) () ( This density is otained fro zero-ean circular Gaussian statistics y a siple transforation of variales. Notice that an assuption is ade that data points are statistically independent. Thus, instead of counting pixels, Sith and Wagner chose to deal with ultrasonic speckle spots, which have diensions defined y correlation lengths. Statistical decision theory was then put to use. The prole they were interested in was the task of target detection, where the target was for =

3 exaple, a disk signal represented y an increased region of variance. Signal present and signal asent variances ay e denoted as ψ and ψ -. The test statistic of the ideal oserver is the log-likelihood ratio pdf ( v ψ ) λ( v) = log () pdf ( v ψ ) Here pdf ( v ψ ) and pdf ( v ψ ) are the likelihoods (proailities) of the signal eing present and asent respectively. Ignoring irrelevant constants and scaling factors, this test statistic is shown to e λ(v) = N v = Using first and second oents of the Rayleigh distriution, the ideal oserver lesion detection signal-to-noise ratio is given as I [ λ λ ] [ σ σ ]/ = (4) = [ ψ ψ ] A I N = ψ ψ S c o C (3). (5) λ and σ are the ean and variance of the test statistic λ assuing the lesion is present, and λ and σ are the ean and variance assuing the signal is asent. A is the lesion area, Sc is the area of a speckle spot, o is the pixel signal to noise ratio (.9 for Rayleigh distriution), C is the contrast of the lesion, and N is the nuer of independent saples (speckle spots). The speckle spot area is the integrated noralized autocovariance of the iage field, and depends on the syste design. When there is a correspondence etween spatial resolution and speckle spot size, the equation tells us that detectaility is etter for high-resolution systes ecause there is ore independent inforation availale for aking decisions. Task perforance is also iproved with increasing lesion contrast. The Sith-Wagner theory was a reakthrough for ultrasonography ecause it connected for the first tie it estalished a rigorous connection etween engineering properties that we easure in the laoratory and clinical task perforance. Conveniently, syste properties could e uncoupled fro oject properties. The Sith-Wager theory 0 used soe stringent assuptions including the following: () Electronic noise was neglected in the analysis. () The theory used envelope detected signals instead of the raw RF data. Further insights into optial detection ight e availale y using unprocessed data. (3) Speckle spots instead of pixels were used. A odified theory that allowed for correlated pixels would e ore satisfying. (4) There was an iplicit assuption ade that only large lesions were eing considered, as the variance for oth hypotheses were constant across iage data. Hence clutter due to ea sideloes, and related edge effects are not descried in this theory. A detection theory that addresses saller lesions ay prove very iportant. (5) An assuption of linear shift-invariance was ade. We propose a theory that will extend the Sith-Wagner theory, and relax soe of the assuptions upon which it is ased. To address these questions we turn to the toolox of literature availale for iage quality assessent. In particular, three papers on ojective assessent of iage quality y Barrett and colleagues 3,4, provide us with a rigorous foundation on which to ake further developents. These papers deal with estiation and detection task perforance assessent. They ake connections to the Rose theory of contrast and the Wagner-Brown odel for assessent of detection tasks LINEAR SYSTEM MODELS FOR ULTRASOUND SIGNALS Ultrasound systes transit acoustic waves into tissue and for iages fro the scattered signals that return to the transducer. Ultrasound radio-frequency (RF) echo signals can e odeled as a shift-variant linear syste of the for 4 g ( t) = dxh( x, t) γ ( x) n( t). (6)

4 Here g is the voltage trace echo signal, n is the noise, and h is the iaging syste response to a delta function oject. h represents all (linear) aspects of the syste including the transit wavefor, electroechanical coupling of the transducer, diffractive propagation, scattering, reception, and eaforing. It does not include nonlinear processing steps such as envelope detection. γ is the oject function, which is represents spatial perturations in density and copressiility. Unlike other odalities, ultrasound systes are not sensitive to the shape or distriution of hoogeneous aterials ut rather spatial fluctuations in ediu properties. Consequently, a scattering ediu is ost usefully characterized statistically. The variance of γ is associated with echo ackscatter strength. The spatial vector x defines the location in the oject space, and the teporal vector t spans the data space. For a single A-scan line, t ay e a -D vector. The echo data for a -D B-ode iage uses a -D atrix for t. For higher diensional acquisitions, e.g., flow and elastography, t further increases in diension. For discrete iages of M pixels the linear syste can also e written in vector-operator notation as: g = H γ n (7) where H is a continuous-to discrete operator and the data g is now a vector or atrix with M eleents. 4. IDEAL OBSERVERS OF THE RF SIGNAL It is the statistical properties of the echo signal that give us inforation aout the oject iaged. The proaility density functions for signal present and signal asent RF echo signals can e odeled as zero-ean ultivariate noral densities t exp g K g ( g ) = t exp g K g pdf and pdf ( g ) =. (8) M / M / π det K π det K ( ) t Here the covariance atrices = gg t K and K = gg ( ) descrie the iage textures for the signal present and signal asent hypotheses respectively. A lesion present, thus, for exaple, ay have a region of increased variance relative to the surrounding tissue. The log-likelihood ratio is given as pdf ( g ) t λ ( g) = log = g [ K K ] g [ ( K K g log det )]. (9) pdf ( ) We can ignore the second ter as it is a data-independent constant. The ideal oserver test statistic gives the strategy for optial detection. The strategy is to copute the difference etween the squared agnitudes of the pre-whitened data. In section 8 we coent on the iportance of the ideal oserver strategy for iproving huan oserver perforance y iage processing. Eq. 9 represents a signal known statistically (SKS) quadratic task. By quadratic task we ean the test statistic is a quadratic for of the data vector g. At this point we can copute the of the test statistic 5 using (4): { tr[ ( K K )( K K )]} [( K K ) K ] tr [ K K K ] { tr[ ] [( ) ]} λ = (0) 8 where tr[. ] is the trace of the atrix. The expression in the denoinator ade use of Isserlis s forula for fourth order oents 6 of Gaussian distriuted signals. Eq. 0 descries how echo signal textures due to the syste and oject interrelate to affect task perforance. It is iportant ecause it extends the Sith-Wagner theory to include pixel correlations, and in fact the reduces to the Sith-Wagner odel when covariances are diagonal and stationary, and pixels are sapled on the sae scale as correlation lengths. It is not a very intuitive odel, however, and could e coputationally expensive to evaluate due to the inverse covariances needed. Our ultiate goal is to gain ore intuition of ideal oserver perforance and so we consider how other approaches could copleent this analysis. 5. CLARKSON-BARRETT APPROACH Here we suarize an approach to analysis taken y Clarkson and Barrett 7. One contriution of our paper is to point out the applicaility of the Clarkson-Barrett theory to ultrasound perforance assessent. We shall also use this theory later on to understand a ore intuitive picture of iage quality.

5 The ideal oserver is dependent on the proaility distriutions of the test statistic conditioned on the hypothesis that the signal is present or asent. Alternatively, instead of pdf s, oent-generating functions for the log-likelihood ay e used. They are essentially the Laplace transfor of the pdf s, M ( β ) = exp βλ and M ( β ) exp βλ () ( ) ( ) = and hence are also related to the characteristic functions (Fourier transfor) of the pdf s (likelihoods). Because proaility ust e conserved (proaility of lesion present proaility of lesion asent = ), all the inforation aout the area under the ROC curve is contained in one of the oent-generating functions. The oent-generating functions can consequently e written in ters of a single function, called the likelihood generating function G(β): M ( β ) exp β β G β / and M ( β ) = exp β β G / () [ ( ) ( )] [ ( ) ( )] = β Both oent-generating functions and likelihood generating functions are useful for quantifying detectaility of the signal with stochastic calculations. Different kinds of etrics can e used to quantify detectaility. We have already discussed λ. The area under the ROC curve (AUC) is also a useful easure of detectaility. Clarkson and Barrett show that λ has the unusual ehavior that it does not increase without ound as signal strength increases. This is also not invariant to onotonic transforation of the decision variale, whereas the AUC is. Another is given y [ G(0 ] / G =, (3) ( 0) ) where G ( 0) = 4log M (/ ). (4) This is invariant under onotonic transforations of the decision variale, and is related to the Bhattacharyya distance, a etric quantifying the distance etween two general proaility density functions p (x) and p (x): / d B ( p, p ) = log dx[ p( x) p ( x) ]. (5) When the two distriutions have no overlap, there is very good separaility. In this case, the integral tends to zero, and the Bhattacharyya distance ecoes infinite. When there is coplete overlap, the distriutions are identical and noralization constrains the integral to. Consequently the Bhattacharyya distance tends to zero. For our task, the essential oent generating function is β det K det K M ( β ) = (6) det [( β ) K βk ] and the corresponding signal-to-noise ratio is ( ( )) det K K G( 0) = log. (7) N det K det K Intriguingly and conveniently this does not require inversion of any covariance atrices. It s coputational evaluation ay nevertheless e prohiitive for non-stationary covariance atrices due to expensive deterinant operations. 6. ADDITIVE SIGNAL HOTELLING (IDEAL) OBSERVER One unsatisfying aspect of the derived figures of erit for the quadratic task of detection is that the iaging syste paraeters are uried inside covariance atrices. Hence interpretation and intuition for syste optiization is not ovious. Moreover, for the proles we are interested in, the oject statistics are non-stationary (statistical properties of the iage are location-dependent), hence the echo signals theselves are non-stationary. The advantage of stationarity is that diagonalization is natural and algorithically efficient with a discrete Karhunen-Loeve expansion. Lacking such stationarity, coputational evaluation of iage quality etrics can e prohiitive. Another approach to odeling the signal detection prole is as follows. Consider an oject function for the signal asent that is a zero-ean stationary stochastic process γ (x) with covariance K. For the signal present case consider β

6 that the oject function can e odeled as the rando ackground γ (x) plus an additive signal γ(x) that is one deterinistic realization of a rando process (the statistical properties of this rando process ay vary fro point to point in space). In essence, we are using the concept that a ultiplicative signal in the variance can e represented as an additive signal. The advantage of this is that we are now ack to a prole that is well known in the iage quality literature: an additive signal s = H{ γ(x)} and stationary covariance K=H t K HK n, where K n is the noise covariance. The prole is descriptive of hyper-echoic lesions (lesions where the ackscattered intensity is greater than the surrounding tissue ackground). The optiu linear oserver is the Hotelling oserver 3 which has test statistic s K g (8) with given y λ Hot = t [ ] = tr[ K ] ss t Hot. (9) It has een shown that this is identical to the ideal oserver for non-rando signals in norally distriuted rando ackgrounds. The strategy for the Hotelling oserver is to perfor atched filtering with a pre- whitening step. Assuing local shift-invariance, K is stationary, and a frequency space description of the is possile via a Fourier transfor. Additionally, we average over signal realizations to otain: I γ Γ( u) = du H ) S γ H ) γ. (0) ) S ) Siilar averaging of has een done efore except over location uncertainty. Here Γ(u) is the Fourier description of the oject function signal γ(x), S γ (u) is the power spectru of the ackground oject function process, and S n (u) is the noise power spectru. H(u) is the syste response function. The integration is done out to the extent of the sapling frequency. This is an ensele average over a generalized Wagner-Brown odel of detectaility 6,,3. Before exaining the interesting properties of this, we first exaine the expectation value of the agnitude of the Fourier doain target signal in the nuerator of the integral. Consider that the spatial doain target signal is given y γ ( x) = w( x) ξ ( x), where w is a deterinistic window function and ξ is stochastic process. The purpose of writing γ in this for is to characterize the location dependence of target signal statistical properties. For exaple, w could e a disk signal that is unity inside the lesion and zero outside the lesion. In general, the Fourier doain stochastic average over target signal variations can e written as S * * Γ ) Γ( u) = W )* Ξ( u) = dpdqw( p) W ( q) Ξ( u p) Ξ q) X X X () where W and Ξ are the Fourier transfors of w and ξ respectively, and X is the spatial region of support over the iage (the iage area or volue). At this point, we look at soe siplifying assuptions to copare our results with those of the Sith-Wagner theory. If in particular ξ is a white Gaussian noise (WGN) process with variance σ ξ, then Ξ is also WGN. The WGN assuption is a good one when spatial variations of density and copressiility are on a uch saller scale than the resolution of the iaging syste and saller than the sapling intervals used. With this assuption, S Γ n ) = σ dx w( x), () ξ y Parseval s theore. If, for exaple, we odel disk signals as Sith and Wagner did, such that w(x) is unity where the lesion is present and zero where it is asent, we have S Γ ) = σ ξ A, where A is the area of the lesion (or volue of the lesion if considering 3-diensions). Thus, when the oject statistics can e treated as WGN, S Γ (u) is a constant, that is proportional to the oject variance and the area of the lesion. Hence,

7 GNEQ I γ H ) = σ ξ AX du = σ u ) ξ AX d GNEQ. (3) H ) S ) S ) We have written this quantity as an integration over what has een tered the Generalized Noise-Equivalent Quanta or GNEQ, defined as the integrand of the third ter. Noise-Equivalent Quanta (NEQ) has historical origins with Shaw 7 and others, as well as the Wagner-Brown theory of detectaility 3. For photon iaging systes NEQ represents the frequency-specific density of quanta at the input of an ideal detection syste that would yield the sae output noise as the real syste under evaluation. Generalized NEQ as descried y Barrett and colleagues 6, provides provision for a stochastic ackground texture of the oject. For ultrasound systes the GNEQ quantity is a easure of the spatial frequency sensitivity of detecting a signal in a ackground texture and in the presence of electronic noise. For photon iaging odalities, the ideal oserver detectaility is given y an integral over the frequency-doain of NEQ (or GNEQ) ties the squared agnitude of the Fourier transfor of the signal. For ultrasound systes, the GNEQ is not weighted y the agnitude of the Fourier transfor of the signal shape, ut rather y the spectral variance of the target signal. This is to e expected since ultrasound systes are not sensitive to the agnitude of density or copressiility ut rather depend on the variance (spatial fluctuations) of these quantities. The entire GNEQ spectru can e used to characterize ultrasound systes for a particular tissue type in a target-independent anner. Defining n as the integrated ackground power spectral density, we can also re-write the as: MTF ) GNEQ = Coj A du, (4) S ) γ MTF ) X S / N ) σ / σ where C oj is the oject function contrast, defined as C oj = σ ξ and MTF is the odulation transfer function defined as MTF ) = H ) / H (5) where H ax is the axiu value of H(u). The ackground signal to noise ratio is defined as: S / N ax σ ) = H Xσ / S ), (6) ax and is a easure of the ackscattered signal strength of the oject ediu relative to the noise level. Note the connections etween our expression for and the Sith-Wagner theory: oth easures are proportional to lesion area. Curiously, [ GNEQ ] is proportional to contrast, yet in the Sith-Wagner theory, the λ is proportional to the square of oject contrast. Both etrics, however, reveal that task perforance is iproved with iproved contrast. One elegant feature of this foralis is the direct dependence of syste and noise properties on the detectaility. Also, the description is in the Fourier doain, a natural choice for analyzing ultrasound iaging systes ecause focused fields in the Fresnel region and unfocused apertures in the far-field (Fraunhoffer) region have k-space descriptions equivalent to the Fourier transfor of the aperture. Also, the quantities involved in the GNEQ(u) are all easurale quantities, so this could provide a convenient way to characterize a syste design experientally! To do so we would need to easure the MTF(u) and the noralized ackground plus noise power spectru (NBNPS), which we define as: NBNPS ) H ) S ) S ) / H, (7) ( n ) ax then take the ratio of the two: GNEQ ) = MTF( u) / NBNPS( u). (8) Wire or ead phantos can e used to estiate the point-spread functions and consequently the MTF. It ay e necessary to acquire a sall nuer of iages to average over noise properties. Assuing local ergodicity, a patch of one or a few (RF) iages ay e used to estiate the ackgroundnoise power spectru for a particular field region. Once we know these properties, the integrated GNEQ or IGNEQ as we shall call it ay e coputed: IGNEQ n dugneq(u = ). (9)

8 This etric ay apply to edia where oject statistics can e treated as nearly white, such as standard phantos and any types of tissues. Conveniently, this etric is independent of the signal size and contrast, and characterizes the perforance of the iaging syste with a single nuer. It ay e noralized y to otain a nuer useful for siulation studies, which we shall call the noralized IGNEQ. Tissues that have coplex striations or variale patches of rightness ay ake target discriination ore difficult. Oject texture can e accounted for in the theory and easured experientally. σ 7. MONOTONICITY WITH THE FULL QUADRATIC TASK Recall that when we odeled oth the lesion and its surroundings as Gaussian stochastic processes the resulting ideal oserver test statistic was a quadratic for in the data. In contrast, when we odeled the lesion as an additive deterinistic signal, the test statistic was linear in the data. We ay very well expect that perforance etrics for these two detection tasks ay not exhiit identical properties. Consequently, we now exaine whether the figures of erit derived for oth tasks give siilar inforation aout diagnostic perforance. To do so we choose to exaine the trends of oth the Clarkson-Barrett and the GNEQ theory with varying syste and oject paraeters. Consider the case where oth K and K - are siultaneously diagonalizale with a Karhunen-Loeve transforation. In assuing this we consider the case where the size of the lesion is the sae size as the window [w(x)= everywhere], and also that the syste is locally shift-invariant. The result is a spectral description: M [ ( ) ] H ) S ) S ) Sn ) = G( o) = log M M, (30) M [ ] [ ] H ) S ) Sn ) H ) S ) Sn ) = = where S (u) and S (u) are the eigenvalues (spectru) of K and K - respectively, and u is a (spatial) frequency vector of length M. Note that S ) = S ) and S ) = S ) S ), where ) is the spectru of the ackground oject function, and (u) is the spectru of the target. Also, (u) is the noise-power spectru (NPS). S γ The spectral description now gives a eans for coparing with GNEQ. In particular, we shall exaine trends of oth G(0) and GNEQ for paraeters of andwidth, noise power, and target signal power. To investigate the ehavior of the with syste andwidth (BW), consider for siplicity an ideal low-pass syste response: 0 u u H ) =. (3) 0 otherwise Suppose also that oject and noise statistics are white: S (u) = S, S (u) = S, and S n (u) = Sn for all u. Then [( ) ] [ ] S S Sn Sn G ( o) = log ( M )log [ ][ ] [ ][ ]. (3) S Sn S Sn Sn Sn The second ter vanishes since the log of unity is zero. Consequently, the is proportional to, which is siply proportional to u, the syste andwidth. To show that G(0) decreases with increasing noise, we took the derivative with respect to the noise power and showed that the result was a strictly negative quantity, indicating that the slope of the graph of G(0) with respect to noise was negative (calculation not shown). As the noise grows larger without ound, oth G(0) and GNEQ oth tend to zero. As noise power tends to zero, G(0) and GNEQ increase to a liit that is only ound y the density of sapling, and the Nyquist liit of H. If H is non-zero for frequency channels u Ω, then as noise tends to zero: γ S n S

9 H ) du = noise 0 Ω H ) S ) Ω GNEQ du. (33) S ) As the sapling frequency grows, so does the as long as H has even a sall aount of signal. The sae ehavior is seen with G(0) : [ ] Ω [ S ) S ) G( 0) log noise. (34) 0 u S ][ ] ) S ) As the nuer of frequency channels in Ω increases, this quantity will in general increase until oth target and ackground oject power spectra tend to the sae nuer, for exaple, as they die off to zero. In the Sith-Wagner theory, noise-power could e totally neglected, and the detectaility was ounded not y the sapling density, ut y the nuer of independent speckle spots. By using the RF signal instead of the envelope signal, the ideal oserver has the potential to copensate for the lurring kernel of the iaging syste. Noise prevents the ideal oserver fro doing this copletely. One interpretation of the ideal oserver strategy is that it deconvolves the effects of the iaging syste efore perforing teplate atching. Deconvolution cannot necessarily recover precise inforation lost in the nullspace ut it can restore iage statistics useful for the detection task. An interesting oservation related to deconvolution is that the arguent of the GNEQ integral looks very uch like a Weiner filter. In practice, there will always e noise, and the ideal oserver detectaility will e ounded. The iaging syste should e designed so as to saple at iniu at the Nyquist rate, deterined y the syste s MTF. Sapling ore densely than the Nyquist rate will add noise power to the iages (which ay degrade task perforance for huan oservers). For ideal oservers, sapling at greater than the Nyquist liit will not degrade perforance, ut will not provide any enefit either. This is ecause the ideal oserver will know to truncate spectral noise where there is no signal. Lastly, we note the increasing trends of oth G(0) and GNEQ with increasing signal strength. GNEQ is proportional to target signal variance. For G(0) the relationship with target strength is ore sutle, however, y differentiating with respect to signal strength, one can show that the relevant slope is positive (calculation not shown). The conclusion of the previous analysis is that GNEQ, a figure of erit for a slightly different task than the full quadratic task odeled in sections II, IV, and V gives siilar inforation as do etrics for the quadratic task. The enefits are that sall targets (lesions) can e considered, noise and syste paraeters can e included directly and intuitively into the analysis, and the can e oth easily easured and predicted with a target-independent etric. 8. SHIFT-VARIANT INFORMATION MAPS OF THE ULTRASONIC FIELD Realistic ultrasound systes are not shift-invariant. But we can assue that they are shift-invariant over soe local region. By doing so, we can copute the potential inforation content of iaging a lesion at each field location. We considered a 8-eleent linear array transducer with 64 active eleents and fixed focus at 60. No aperture growth, dynaic receive focusing, elevation focusing, or apodization were eployed. Attenuation was neglected, and we only consider diensions of a truly 3-D ultrasound ea sensitivity. All such effects could easily e integrated into future siulations. Point spread functions (psf s) siulated using FIELD II 8, a pulic doain ultrasound siulator, are shown elow in Fig. (a). This is a B-ode iage of point scatterers located at successive axial depths separated y 5 intervals. At each point the MTF was coputed, corresponding to the noralized -D FFT of the RF point-spread functions. Assuing white ackground and noise processes, and S/N = /30 at the focus, a noralized IGNEQ value was coputed at each axial point, and the results plotted in Fig. (). The results are soewhat counter-intuitive. We would have expected the focal region to e optially inforative since it has the ost focused resolution size. Instead we find that the nearfield has rearkaly ore inforation potentially availale. To understand this unexpected result, we copared the nearfield and focal region psf s and MTF s, as shown in Fig.. Intuition tells us that (a) has etter resolution, ut upon exaining the spatial frequency doain, we need to give the question soe ore thought. The effective lateral spatial andwidth in (d) is 74% greater than in (). Thus the lateral spatial resolution of the large curved wavefront (c) is potentially greater than the lateral resolution of the saller focal region field (a)! To understand this etter, notice that a lateral slice through the psfs looks very uch like a chirp

10 (a) () lateral distance () axial distance () IGNEQ axial position () Figure. (a) Point spread functions due to 64 active eleents of a 8-eleent linear array transducer of height 5, eleent width λ, and gap spacing of 0., with fixed focus at 60, and no elevation lens. An attenuationless ediu was considered for siplicity. cycles of a 3 MHz sinusoid weighted y a Hanning window were used to siulate the excitation pulse. The sae function was used to siulate the electroechanical coupling ipulse response of the transducer. () The noralized IGNEQ values corresponding to field points along the ea axis. The larger the noralized IGNEQ value the ore inforative the syste is for the given field point. (a) 58 z [] (c) x [] 8 z [] x [] () -0 kz [cycles/] kz [cycles/] 0 0 (d) kx [cycles/] kx [cycles/] Figure. (a) Focal and (c) nearfield RF psf s due to a 3 MHz fixed focus linear array of height 0.5 c, with 64 active eleents of width λ separated y distances of 0.. Aziuthal focus was 6 c. The ea was electronically swept laterally across the point target located at (a) 6 c and () 3 c. () and (d) are the k-space (MTF) representations of (a) and (c) respectively. function. The resolution ust e recovered y processing siilar to current coded excitation schees 9. Siilar to Jensen 0 we propose a atched filter technique, wherey a tie-reversed copy of an RF psf (c) is used as a filter. When convolved with an RF iage of a point target (c), a uch narrower lateral distriution 3() is the result. The signal

11 aplitude in 3() is also ore than 0 ties greater than that in 3(a), and. ties greater than the focal gain. A arked iproveent in visiility is seen when atched filtering is applied to siulated iages of a cyst phanto, shown in Figs. 3(c) and (d). The atched filtering procedure is consistent with the strategy for the ideal or Hotelling oserver, which is to whiten and atch filter. In the nearfield, atched filtering without whitening perfors well ecause the resulting iage statistics are nearly white. It reains to e seen how roust the technique is to phase aerrations and other artifacts. Regardless of whether spatial atched filtering is a practical technique, it shows an iportant point that iage processing can ake inforation in the raw RF data ore accessile to huan oservers, as seen in Fig. 3(d). Processing does not, however, increase the inforation content. The raw RF signal contains all the task-relevant data. Processing can at est aintain inforation content, and often, inforation will e lost. We should coent that eaforing schees (at least for reception) can e viewed as either part of the syste design or as part of the processing algoriths. Both views ay have erit, and further research in this area could e pursued. Before concluding, we coent on the general downward trend of detectaility with increasing axial distance as shown in Fig. (). Diffractive losses ean that the farfield has less signal availale than does the nearfield, and consequently the S/N decreases, dropping the overall detectaility. (a) 8 z [] (c) x [] z [] x [] () (d) x [] x [] Figure 3. (a) The envelope-detected psf in (c). () The resulting iage otained after atched filtering (c) with a tie-reversed replica filter. (c) Pre-filtered iage of a lesion due to psf (c). (d) The corresponding post-filtered iage. 9. DISCUSSION AND CONCLUSIONS One result of our analysis is an expression for the ideal oserver detectaility that is a generalization of Noise-Equivalent Quanta (NEQ), a easure widely used y other iaging odalities. The detection signal to noise ratio is shown to e an integration of the generalized NEQ weighted y the spectral variance of the target signal (rather than the squared agnitude of the Fourier transfor of the signal, as is the case with other odalities). This reflects that ultrasound systes are sensitive not to the agnitude of the ediu paraeters ut rather depend on the variance (spatial fluctuations) of these quantities. Moreover, when oject statistics are white over the sapling doain, the frequencydependent weighting of GNEQ ecoes constant, and a target-independent picture of detectaility can e suarized in a single nuer. The resulting fraework ehaves siilarly to the quadratic task, is intuitive, and is aenale to easureent and prediction of syste perforance. The theory has een used to predict the inforation content of the ultrasonic ea at various field points. Siilar analysis of location-dependent IGNEQ ay prove useful for evaluating

12 the effectiveness of eaforing strategies. Currently lesion detectaility is assessed with tie-consuing Monte-Carlo siulations, or y using phantos with real systes. Backscattered power inside and outside the lesion are easured in ratio to give a detectaility etric. The figures of erit presented here are instead ased on statistical odels and have a different intention: they give an upper ound on the inforation content availale with a particular syste design. The theory does not attept to predict huan oserver perforance. However, new strategies iicking the ideal oserver are revealed for processing RF data that could iprove detection of lesions for huan oservers. Experiental characterization of ultrasound systes is planned in the near future. ACKNOWLEDGEMENTS This work was supported in part y the National Institutes of Health R0 CA REFERENCES T. M. Kol, J. Lichy, J. H. Newhouse, Coparison of the perforance of screening aography, physical exaination, and reast US and evaluation of factors that influence the: an analysis of 7,85 patient evaluations. Radiology, 5, pp , 00. T. M. Kol, J. Lichy, J. H. Newhouse, Occult cancer in woen with dense reasts: detection with screening US-- diagnostic yield and tuor characteristics, Radiology, 07, pp. 9-99, H. H. Barrett, Ojective assessent of iage quality: effects of quantu noise and oject variaility, J. Opt. Soc. A. A, 7, pp , H. H. Barrett, C. K. Aey, and E. Clarkson, Ojective assessent of iage quality. III. ROC etrics, ideal oservers, and likelihood-generating functions, J. Opt. Soc. A. A, 5, pp , H. Van Trees, Detection, Estiation, and Modulation Theory, New York, Wiley, 97 6 H. H. Barrett, J. L. Denny, H. C. Gifford, and C. K. Aey, Generalized NEQ: Fourier analysis where you would least expect to find it, SPIE 708, pp R. Shaw, The equivalent quantu efficiency of the photographic process, J. Photog. Sci., pp , R. F. Wagner, S. W. Sith, J. M. Sandrik, and H. Lopez, Statistics of speckle in ultrasound B-scans, IEEE Trans. Son. Ultrason., 30, pp , R. F. Wagner, M. F. Insana, and D. G. Brown, Statistical properties of radio-frequency and envelope-detected signals with applications to edical ultrasound, J. Opt. Soc. A. A, 4, pp. 90-9, S. W. Sith, R. F. Wagner, J. M. Sandrik, and H. Lopez, Low contrast detectaility and contrast/detail analysis in edical ultrasound, IEEE Trans. Son. Ultrason., 30, pp , May 983. H. H. Barrett, J. L. Denny, R. F. Wagner, and K. J. Meyers, Ojective assessent of iage quality. II. Fisher inforation, Fourier crosstalk, and figures of erit for task perforance, J. Opt. Soc. A. A,, pp , 995. A. E. Burgess, The Rose odel, revisited, J. Opt. Soc. A. A., 6, pp , R. F. Wagner and D. G. Brown, Unified analysis of edical iaging systes, Phys. Med. Biol. 30, pp , R. J. Zep, C. K. Aey, M. F. Insana, Linear syste odels of ultrasound iaging: Application to signal statistics, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., To e pulished. 5 R. J. Zep, C. K. Aey, M. F. Insana, Fundaental perforance etrics and optial iage processing strategies for ultrasound systes, Proc. IEEE Ultrason. Syp., Munich, Gerany D. Middleton, An Introduction to Statistical Counication Theory, pp. 343, Penninsula Pulishing, Los Altos, CA, E. Clarkson and H. H. Barrett, Approxiations to ideal-oserver perforance on signal-detection tasks, Applied Optics, 39, pp , J. A. Jensen and N. B. Svendsen, Calculation of pressure fields fro aritrarily shaped, apodized, and excited ultrasound transducers, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., 39, pp. 6-67, M. O Donnell, Coded Excitation Syste for Iproving the Penetration of Real-Tie Phased-Array Iaging Systes, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., 39, pp , J. A. Jensen, P. Gori, Spatial filters for focusing ultrasound iages, Proc. IEEE Ultrason. Syp., Atlanta, USA, pp , 00.