1704 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 49, no. 12, december 2002

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1 1704 ieee tranaction on ultraonic, ferroelectric, and frequency control, vol. 49, no. 12, december 2002 Proceing Radio Frequency Ultraound Image: A Robut Method for Local Spectral Feature Etimation by a Spatially Contrained Parametric Approach Jean-Marie Gorce, Deni Friboulet, Igor Dydenko, Jan D hooge, Member, IEEE, Bart H. Bijnen, Member, IEEE, and Iabelle E. Magnin, Member, IEEE Abtract Spectral etimation i a major component in tudie aiming at characterizing biological tiue through the analyi of backcattered radio frequency (RF) ultraonic ignal and image. However, conventional pectral etimation technique yield a well-known trade-off between patial reolution and variance. The backcattered ignal are tochatic by nature, o hort-term local analyi reult in a high variance of the etimate, which cannot efficiently be reduced through conventional patial averaging. We addre thi iue by decribing a pectral etimation technique that reduce the variance of the etimate (by moothing the local etimate in pectrally homogeneou region) while preerving pectral dicontinuitie (i.e., the moothing i not performed acro region with different pectral content). The propoed approach i et in a Bayeian framework and i baed on local autoregreive (AR) etimation, contrained by moothne prior. Thee moothne prior are introduced through a Markov random field in which the aociated potential function are nonquadratic, allowing thereby to preerve dicontinuity. The method i validated on imulated RF image and teted on echocardiographic image acquired in vivo. The reult are compared to the etimate provided by the conventional Burg technique. Thee reult clearly demontrate the ability of the propoed approach to improve pectral etimation in term of variance reduction and dicontinuity detection. I. Introduction Quantitative analyi of radio frequency (RF) ultraonic ignal backcattered from biological tiue provide the meaure of variou ultraonic parameter that reflect the underlying inonified tiue tructure and biochemical compoition. For intance, normal or infarcted myocardium a well a it fiber tructure ha been aeed uing backcatter meaurement [1] [8]. Mean catterer pacing etimation ha been tudied to differentiate normal and cirrhotic liver [9] [11]. Attenuation and integrated backcatter have hown promie to characterize Manucript received April 17, 2001; accepted July 1, J.-M. Gorce i with CITI, INSA-Lyon, bat. Leonard de Vinci, Villeurbanne Cedex, France ( jean-marie.gorce@ina lyon.fr). D. Friboulet, I. Dydenko, and I. E. Magnin are with CREATIS, CNRS Reearch Unit (UMR 5515), affiliated to INSERM, INSA- Lyon, bat. Blaie Pacal, Villeurbanne Cedex, France. J. D hooge and B. H. Bijnen are with the Medical Imaging Computing, Department of Cardiology, Catholic Univerity Leuven, Leuven, Belgium. atheroclerotic plaque [12] [14]. Mean catterer ize ha been propoed to invetigate the vacular tructure of renal tiue [15], [16]. Thee tudie thu open the perpective of detecting and identifying leion by forming parametric image reflecting the patially varying tiue tructure of the inonified organ, baed on the analyi of a et of RF ignal, i.e., RF image. The contruction of uch parametric image ha been invetigated for repreenting the two-dimenional (2-D) patial ditribution of mean catterer ize in renal tiue [15], [16], of backcatter coefficient [17] [19], of integrated backcatter in myocardium [20] or attenuation [12] in the aortic arterial wall. Spectral etimation i a major component of thee tudie a the 2-D ditribution of the above-mentioned parameter i derived from the computation of the local power pectral denity (PSD) over the RF image. The ue of technique uch a FFT [1], [4], [12], [13], [17], [18], [21] or auto regreive (AR) modeling [9], [10], [15], [22], [23], yield a well-known trade-off between patial reolution and variance. Indeed, hort data window are required to enure a high patial reolution but in turn yield a high variance of the etimation. Thi variance i uually reduced by moothing a poteriori the pectral etimate, i.e., by averaging the local etimate patially [17], [19], [21]. However, patial averaging provide reliable reult only if it i performed over pectrally homogeneou region. Thi repreent a pecific problem for pectral etimation from RF image becaue they are backcattered from an area encompaing different or heterogeneou tiue and their pectral content may thu exhibit large variation (i.e., pectral dicontinuitie) at interface between thee tiue. For intance, in cardiac imaging the inonified tiue (i.e., myocardium and blood cell) include catterer having different hape and ditribution; a a conequence, they exhibit different frequency repone, inducing, in particular, a hift of the apparent pule power and central frequency [5], [22], [23]. Thi problem ha lead ome author to perform averaged pectral etimation in manually delineated region of interet [1], [13], [21]. In order to reduce the variability of attenuation meaurement, Bridal et al. [12] define region of interet baed on the homogeneity of attenuation lope /$10.00 c 2002 IEEE

2 gorce et al.: radio frequency ultraound image and pectral feature etimation 1705 The previou remark call for a method allowing the average of the local pectral etimation in homogeneou region while preerving dicontinuitie between different tiue. In thi paper an approach addreing thi iue i decribed, baed on contrained AR pectral etimation. In the propoed approach, the et of AR parameter to be etimated i conidered a a 2-D random field. A propoed by Kitagawa and Guerh [24], Giovannelli et al. [25], [26] the patial moothne contraint i introduced through a prior cot function in a Bayeian framework [24]. Thee tudie, however, did not deal with the problem of abrupt dicontinuitie between local pectra etimate. Following our previou work [27], [28], and a uggeted in a imilar way by Barbareco [29], [30], we propoe in thi tudy to take thee dicontinuitie into account by expreing the moothne prior through a Markovian random field (MRF). The MRF decription provide a convenient formalim to expre the deired local tatitical propertie of a random field through the definition of appropriate clique and potential function. The ue of nonquadratic potential function enable to preerve dicontinuitie and the maximization of the global poterior probability lead to the maximum a poteriori (MAP) etimate [27] [30]. The MAP etimate i expreed in term of the reflection coefficient aociated with the AR model. Section II and III decribe the theoretical apect of the propoed model. Algorithm and numerical detail are provided in Section IV. The behavior and the accuracy of the propoed approach are aeed through a computer imulation in Section V. Section VI alo dicue the behavior of thi method on real echocardiographic RF image. Promiing reult are obtained howing the robutne of the propoed method. II. AR Etimation A. Conventional Etimation of AR Model An AR model of order p (AR (p) ) i a parametric repreentation of a dicrete ignal y(n) decribed a the output of a linear filter, driven by a white Gauian noie ɛ(n) with zero mean and variance σ 2. The output of the AR filter i given by: y(n) =ɛ(n) p a (p) (k)y(n k), (1) k=1 where a (p) (k), k [1; p] are the AR coefficient, and p i the order of the AR model. The power pectral denity (PSD), namely P yy (f), correponding to thi AR model i known to be: σ 2 P yy (f) = 1+ p. (2) a (p) (k) e j2πfk 2 k=1 The pectral etimation problem thu reduce to the etimation of the AR model parameter, including AR coefficient and input noie variance. Several etimator have been derived for thi purpoe. The mot common i the leat-quare (LS) etimator [31]. In order to perform thi etimation, a recurive method ha been developed [31]. In thi method the AR (p) model i recurively approximated by lower AR model, tarting with order 1 and uing the Levinon-Durbin recurion given by: â (1) (1) = ˆγ (1), â (q) (i) =â (q 1) (i)+ˆγ (q) â (q 1) (q i) i =1, 2,...,q 1 1 <q p, (3) where ˆγ (q) i an etimation of the q th order reflection coefficient, and a i the conjugate of a. The etimation of the reflection coefficient from finite time erie may be performed by uing different technique [31]. The well-known Burg etimation i one of thee approache, which lead to unbiaed olution for true AR ignal and which provide a computationally efficient approach. Thi algorithm i the heart of the propoed contrained approach. It hould be noticed that an AR (p) model i fully decribed a well by the AR coefficient or by the reflection coefficient. B. AR Etimation Applied to RF Image A mentioned in the introduction, the nontationary nature of RF ignal implie the ue of local pectral etimation technique intead of a global approach. The AR modeling previouly decribed may till be ued by plitting RF line in ucceive hort window in which the ignal i aumed to be tationary. Thi aumption i valid if window length are mall enough compared to the rate of nontationaritie. To fulfill thi requirement, the RF data are plit into a et of cell uch a each cell =(i, j) contain a hort RF ignal (typically le than 32 point). Then, the pectral etimation i performed within each cell, and the moothing i applied acro neighboring cell. Such a formalim i convenient to develop the patial moothing in a way imilar to the one ued in retoration image technique (for intance ee [32] [34]). The time-pace (TS) image i thu the et of all hort RF ignal and i given by: Y = {y ; S}, (4) where S i the et of cell, andy i the hort ignal aociated with the cell. The pectral etimation problem conit in finding the AR coefficient for all hort ignal y and may be firt conidered locally. The AR etimation (ee Section II-A) may then be ued to compute the local PSD through (2). A mentioned in the introduction, uch a purely local approach yield a high variance of the pectral etimate. However, averaging a poteriori the obtained pectra over neighboring cell, a propoed in [12], [17], [19], [22], would

3 1706 ieee tranaction on ultraonic, ferroelectric, and frequency control, vol. 49, no. 12, december 2002 decreae the patial reolution and would caue the lo of dicontinuitie. In Section III, a global approach allowing to olve thi trade-off i decribed. In thi approach the reflection coefficient of order q, aociated with each cell, are etimated imultaneouly. Thi etimation i performed recurively tarting at order 1 and ending at order p. The parameter to be etimated at order q are given by: [ X (q) = {x (q) ; S}; x (q) = γ (q) ; σ(q)2 ] t. (5) III. Spatial Smoothing in a Bayeian Approach A. Bayeian AR Etimation The Bayeian framework already ha been ued by Kitagawa and Guerh [24] and Giovannelli [26], repectively, in the field of eimic and acoutic Doppler ignal and for the purpoe of AR etimation from nontationary time erie. In thi ection, their approach i extended for 2-D RF data, and a different moothing contraint i ued. The contraint ued by both author [24], [26] i a quadratic contraint, which implie a trade off between patial dicontinuitie preervation and moothne. The MRF modeling allow to introduce nonquadratic contraint, which yield moothne in homogeneou region while preerving patial dicontinuitie. In thi framework, the poterior log-likelihood of an event X may be written a [32]: U pot (X/Y) U like (Y/X) + U prior (X), (6) where U pot (X/Y), U like (Y/X), U prior (X) are, repectively, the poterior, the likelihood, and the prior cot function. The MAP i the mot popular etimator aociated with Bayeian method and conit in the maximization of the poterior probability (i.e., minimization of the poterior cot function). When uing thi formalim for the recurive etimation of the reflection coefficient, the q th order etimate are given by: X (q) =argmin X (q) (U pot (X (q) /Y, X (1)... X(q 1) )). (7) B. Likelihood Cot Function The hort ignal y are aumed to be tatitically independent, making the global likelihood equal to: U like (Y/X) = U like (y /x ), (8) where U like (y /x ) i the local likelihood, which reflect the probability of having y under the knowledge of x. The local dependency between neighboring ite will be introduced in the next ection by the ue of an appropriate prior cot function. The likelihood of y may be defined uing AR model. In the context of a recurive etimation of the reflection coefficient, the ucceive likelihood cot function have been derived in [35], leading to: ( U like y, x (1)... x (q 1) / ) (q) x = log(σ (q)2 ) + ˆρ(q 1) σ (q)2 ( γ (q) ˆγ (q) 2 +1 ˆγ (q) 2), (9) where ˆρ (q 1) i the initial power etimation and ˆγ (q) i the local etimate of the (q) th order reflection coefficient (in the uual leat-quare ene). The minimization of uch a function in aociation with moothing term i not a trivial iue. Moreover, it i difficult to define a moothing term for reflection coefficient and power term jointly. We rather propoe an alternate minimization approach, omewhat imilar to the method propoed by Kitagawa and Guerh [24]. The previou likelihood function may be plit in two interlaced function. The firt correpond to the likelihood of the reflection coefficient with the local power fixed. The econd correpond to the likelihood of the local power term with the reflection coefficient fixed. After obviou computational arrangement, the two following interlaced likelihood function are obtained: and U like (y, x (1) where π (q) U like (y, x (1)... x (q 1) =logσ (q)2 / γ (q) ) ˆρ(q 1)... x (q 1) / π (q) σ (q)2 ) π (q) γ (q) ˆγ (q) 2, (10) ˆπ (q) 2, (11) i the log-power of the reidual noie. ˆγ (q) i the Burg etimate of γ (q) and i the uncontrained olution. ˆπ (q) i alo the uncontrained olution for logpower term and i given by: ˆπ (q) = log(ˆρ (q 1) ( γ (q) ˆγ (q) 2 +1 ˆγ (q) 2 ). (12) Thee function are highly convenient for computational efficiency a they are quadratic. C. Prior Cot Function The prior cot function U prior (X) i derived under the aumption that hort ignal aociated with neighboring cell have imilar pectral propertie. In other word, the PSD of neighboring cell are expected to be cloe when they correpond to the ame tiue; in thi cae, only low and mooth variation are expected, due to diffraction and attenuation. Thi prior knowledge i formalized through a MRF modeling. The MRF modeling for image retoration wa initiated in the beginning of eightie by Geman and Geman [32] a a mean to repreent image feature through probability function. Aociated with the MAP approach

4 gorce et al.: radio frequency ultraound image and pectral feature etimation 1707 function. The choice of the Φ-function i dicued in Section IV-C, 1. The firt order moothing i appropriate for recovering piecewie contant field. In order to recover piecewie planar field, a econd order moothing hould be preferred [34], [36]. For a econd order moothing, the prior cot function include potential function applied to c C (3) and to c C (4) according to Fig. 1: V c={,t,u} (X) =λ Φ(x 2 x t + x u ), V c={,t,u,v} (X) =λ Φ(x + x t x u x v ), (16) where, t, u, andv tand for all poible clique for both 3rd and 4th order according to Fig. 1(d). D. Choice of a Spectral Ditance Fig. 1. Neighborhood ytem: (a) and (b) how, repectively, the 4 and 12 connexity neighborhood ytem. The clique aociated with thee neighborhood are hown in (c) and (d). They, repectively, are ued to define 1t and 2nd order regularization contraint. in a Bayeian framework, the o-called MAP-MRF etimation advocated by Geman and Geman [32] and other [33], [34], decribe the probability of an event in an image feature pace through it likelihood with repect to the oberved data balanced by prior contraint. The MRF modeling require a neighborhood ytem N that link cell, thu defining a tructure for S: N = {N ; S}, (13) where N i the et of the cell neighboring. Fig. 1 how the uual neighborhood ued, repectively, for 1t and 2nd order moothing when uing a regular lattice. Aociated with the neighborhood, a clique, c C (n), i defined a a et of n cell uch a each cell i neighboring to each other in the ene of N. The local characteritic of the MRF are decribed through potential function applied eparately on each clique. The global cot function i obtained uing the Hammerley-Clifford theorem allowing to write it a the um of all local potential function: U prior (X) = c C (n) V c (X), (14) where V c (X) i the potential function applied on each clique. For a firt order moothing, the potential function are defined on 2nd order clique (ee Fig. 1): V c={,t} (X) =λ Φ(x x t ), (15) where and t tand for adjacent cell. Φ(x) i uually called Φ-function and λ i a weighting parameter allowing to adjut the relative influence of prior and likelihood Ideally, the potential function hould be applied on a conventional pectral ditance [37], i.e., in the cae of a firt order moothing: V c={,t} (X) =λ Φ(P (f) P t (f)). (17) However, uch a criteria lead to nonquadratic expreion with repect to the reflection coefficient, and thu the minimization become a nonconvex problem. We propoe intead the ue of quadratic ditance between reflection coefficient and log-power term given by: ( V c={,t} (Π (q) )=λ (0) Φ V c={,t} (Γ (q) )=λ (q) Φ π (0) π (0) t ( γ (q) γ (q) t ), ). (18) Such a choice i guided by the fact that, for our application, the ultraound (US) pule may be aumed to be cloe to a Gauian pule. For a Gauian pectrum having central frequency f c, tandard deviation σ f,andfora given ampling frequency f, it can indeed be hown that the reflection coefficient γ (q) are analytically related to the pectrum through the following equation [35]: γ (q) = e q(πσ f /f ) 2, phae(γ (q) )= 2πq f f c. (19) In an homogeneou tiue, the pectrum exhibit only low variation (due to attenuation and diffraction), thu yielding low variation of the magnitude and phae of the reflection coefficient, which can thu be efficiently moothed through (19). Thi lat obervation i confirmed through the imulation and experimental reult preented in Section V and VI. E. Dicontinuity Adaptive Smoothing Becaue it weight heavily high difference between adjacent cell, conventional quadratic moothing (i.e., Φ(x) =x 2 ) dratically reduce the variance of the etimation, but in turn yield overmoothed etimate around dicontinuitie.

5 1708 ieee tranaction on ultraonic, ferroelectric, and frequency control, vol. 49, no. 12, december 2002 On the oppoite, dicontinuity adaptive (DA) moothing intend to preerve high variation correponding to phyical dicontinuitie while reducing variation due to noie [34], [36]. Initiated by Geman and Reynold [33] and carried on by Charbonnier et al. [38] and Li [34], the deired propertie of a Φ-function for preerving dicontinuitie have been etablihed. Geman and Reynold [33] have hown that the ue of uch a Φ-function i formally equivalent to define a ocalled line proce. The line proce i an unoberved dual field defined on the neighborhood and aociated clique. At each clique c, correpond a variable b c which i inverely proportional to the probability of having a dicontinuity in thi clique. The dual field may be een a a way to weight a quadratic moothing: the moothing i uppreed when a dicontinuity i preent (b c = 0) and i given more weight a b c increae. Let now the variable b c be gathered in a vector B uch a: B = {b c ; c C (n) }. (20) It may be hown [33], [38] that finding the parameter X minimizing U pot (X/Y) i trictly equivalent to finding B and X minimizing U pot (X, B/Y). The traightforward application of thi framework to perform a recurive AR etimation would, however, require to handle independently multiple dual field: one for each reflection coefficient et Γ (q) = {γ (q) ; S}, and noted a B (q) ; q [1; p], one for the log power term Π (q) = {π (q) ; S}and noted a B (0). In order to handle the detection of dicontinuitie over each tep of the recurive etimation, we therefore propoe to ue a unique global dual field B (g) defined a: { B g = b g c =min[δ(0) b (0) c,...,δ (p) b (p) c ]; c C (n)}. (21) The norm of the minimum i ued, which mean that the level of dicontinuity retained for a particular clique correpond to the higher level detected over all dual field. The parameter δ (q) ; q [0; p] correpond to the caling hyperparameter, decribed with the Φ-function in Section IV-C,1. The global dual field play a crucial role in the recurive etimation cheme a dicued in Section IV- B. conit in the etimation of both log-power term and reflection coefficient for all cell in the TS image. The etimation i performed tep by tep from the 0th order up to a deired order p. At each order, two likelihood function have been defined according to (10) and (11). They have to be minimized in aociation with appropriate moothing function according to (14). For a 1t order moothing the poterior cot function are: and U pot (Γ (q) /Y) = S U pot (Π (q)/ Y) = S ˆρ (q 1) σ (q)2 + λ (q) π (q) + λ (0) γ (q) c C ( 2) ˆπ (q) 2 c C (2) Φ ˆγ (q) 2 ( Φ γ (q) ( π (q) ) γ (q) t, (22) ) π (q) t. (23) The minimization conit in minimizing alternately the 1t cot function (22) with Π (q) fixed and the 2nd cot function with (23) Γ (q) fixed. Such a technique allow to reduce the complexity of the cot function to be minimized at each tep. The o obtained likelihood function are quadratic and, therefore, the etimation tak i equivalent to ucceive image retoration problem. The overall tructure of the etimation algorithm i given a: Initialization :ˆπ (0) =logˆρ (0) ; S Evaluate : Π (0) =argminu pot (Π (0) /Y) Π (0) For q =1...p Π (q) (q 1) = Π (primary etimation) Do (alternate etimation) Evaluate Γ (q) =argminu pot (Γ (q) /Y) Γ (q) Evaluate Π (q) =argminu pot (Π (q) /Y) Π (q) Until convergence ( Π (q) and Γ (q) table.) Our experiment indicate that the convergence of the alternate etimation loop (Do... Until) i obtained within one or two iteration, when the primary etimate of power term Π (0) i not too noiy. IV. Algorithm A. Recurive Etimation Scheme The word recurive i ued with repect to the etimation of the reflection coefficient. The etimation cheme B. Minimization Algorithm The etimation of Γ (q) and Π (q) i performed uing the half-quadratic algorithm propoed by Charbonnier et al. [38]. In thi approach, the line proce i explicitly ued,

6 gorce et al.: radio frequency ultraound image and pectral feature etimation 1709 and a minimization i alternately performed with repect to the unknown field X and the dual field B a follow: Initialization : X 0 Do (alternate etimation) Evaluate : B =argmin U pot( X, B/Y) B Evaluate : X =argmin U pot(x, B/Y) X Until convergence, where X tand for either Γ (q) or Π (q), q [0;...; p]. The algorithm i modified in order to take into account the global dual field defined in (21). The evaluation of the dual field B i replaced by a global evaluation leading to: Initialization : X 0 Do (alternate etimation) Evaluate : B =min { Bg δ, arg min U pot( X, B/Y) B Evaluate : X =argmin U pot(x, B/Y) X Until convergence B g = δ B. Thi global dual field introduced in the recurive cheme allow to exploit a dicontinuity detected at a given AR order for the higher AR order coefficient. Thu, the moothing i not performed for clique on which a dicontinuity ha been detected at previou order. For the other clique, a dicontinuity i checked and ued at higher order, if et. Moreover, the ame i done alternately between reflection coefficient and log-power term; at each order, if a dicontinuity i detected for one of them, the dicontinuity i exploited for the etimation of the other one. X i computed with a conjugate gradient algorithm. Thi may be done becaue the poterior cot function U pot (X, B/Y) i quadratic with repect to X when B i fixed. B may be computed analytically [38] by: b c = Φ (u) 2 u, (24) where u tand for x x t in the cae of a 1t order clique. C. Setting and Hyperparameter 1. Φ-Function: Coniderable literature i dedicated to the choice of an appropriate Φ-function. We give herein the broad outline of uch a choice and we refer to [35] for more detail. The ue of a Φ-function having a linear aymptote toward infinity yield a convex poterior cot function (when the likelihood term i convex) having a unique minimum that can be reached uing determinitic algorithm. However, thi choice lead to a biaed etimation [34], [38]. } Fig. 2. Viualization of the propoed family of Φ-function. For n =0, the Φ-function ha a linear aymptote and i equal to that propoed by Charbonnier et al. [38] For n = 1, the Φ-function i bounded and i equal to that propoed by Geman and McClure [39]. In thi repreentation, the value of ɛ i choen to be 10 time lower than δ. On the oppoite, the ue of a bounded Φ-function may lead to an unbiaed etimation, but reult in a nonconvex poterior cot function; therefore, the minimum cannot be guaranteed to be reached when uing a determinitic algorithm. Although the olution may theoretically be obtained in thi cae uing a tochatic algorithm (uch a imulated annealing), the computational burden aociated with uch technique make it difficult to ue in the context of our application in which ucceive minimization are performed. In thi context, a determinitic uboptimal cheme called graduated nonconvexity (GNC) [36] i propoed. The algorithm i baed on the ue of a erie of Φ-function, driven by the variable n (Fig. 2). The firt Φ-function of thi erie ha a linear aymptote, which allow a fat convergence toward a unique initial olution through a determinitic algorithm (i.e., emiquadratic algorithm in our cae). A n increae, the ucceive Φ-function become bounded, yielding unbiaed etimate. In our application, the erie of Φ-function i illutrated in Fig. 2 and defined a: v 2 Φ (n) (u) = [ 1+( δ vδ ) 2n 2 ] 1 n where v = u 2 + ɛ 2 and φ 0 = ɛ 2 [ δ 2n +ɛ 2n 2 φ 0, (25) ] 1 n. δ i a caling factor allowing to adapt the level of dicontinuitie to be recovered. ɛ i ued to enure the continuity of the derivative around 0. In fact, ɛ determine a level under which the moothing become quadratic, and in thi way fixe the level of the accuracy of the reult. It hould be noted that Φ (0) (u) i equal to the Φ- function propoed by Charbonnier [38], and Φ (1) (u) ithe

7 1710 ieee tranaction on ultraonic, ferroelectric, and frequency control, vol. 49, no. 12, december 2002 bounded Φ-function propoed by Geman and McClure [39] for n =1. 2. Smoothing Parameter: The quality of the moothing preerving dicontinuitie depend on everal parameter. Once the Φ-function i choen, λ, δ, andɛ have to be determined for each et of coefficient; ɛ i defined with repect to the expected accuracy of the reult. For the log-variance term Π (0), ɛ (0) i et to Thi yield an accuracy of 0.01 db. δ hould be lower than the dicontinuitie to be recovered. It i aumed that ignificant dicontinuitie hould not be lower than 0.1 db and, a a conequence, δ (0) =0.05 i fixed. The complex reflection coefficient Γ (p) are contained in the unit circle. Becaue the earch domain i bounded, the parameter ɛ i fixed to ɛ (q) =0.001; q [1; p]. In addition, it i aumed that only a ditance between two reflection coefficient higher than 0.01 i ignificant and then δ (q) =0.01; q [1; p] ifixed. The moothing parameter λ i fixed according to the theory of Geman and Reynold [33], for convex Φ-function. They have hown that a limit value for thi parameter may be related to the tandard deviation of the degradation noie η by: λ 3 td(η), (26) where td(η) i the tandard deviation of the additive noie in the degradation model. In our etimation cheme, each minimization tep olved under the Bayeian theory i equivalent to the retoration of a field of unknown corrupted with a white Gauian noie. Therefore, an etimation of the tandard deviation of the degradation noie i required. Let u firt conider the problem of the etimation of the logarithmic AR model input noie variance Π (p).the output of the whitening AR filter applied to each local ignal y i aumed to be a white Gauian noie. For a window length N higher than 8, the tandard deviation of the empirical etimate of the log-variance π (p) may be approximated [35] by: td(ˆπ (p) )= 2 N. (27) Combining thi equation with (26), we find: 2 λ (0) 3 N. (28) Concerning the etimation of the reflection coefficient, the tandard deviation of the etimation of the p th order reflection coefficient of a p th order AR model i approximated by [31]: γ (k) 2 td(ˆγ (p) )= 1 γ(p). (29) N Thi relation i not true for other reflection coefficient, k < p. However, the Burg algorithm i built on the aumption, at each order k, thatk i the order of the AR model. For thi reaon, and becaue of the lack of a tractable general expreion, we ue thi approximation at each tep of the algorithm. Alo, the modulu of the reflection coefficient i not known prior to the etimation and a uboptimal limit i ued provided from (29) by: td(ˆγ (p) )= 1. (30) N Combining thi equation with (26), we find: λ (q) 3 N. (31) follow a Rayleigh ditribution for all k lower than p [31], [35] and the SNR of a Rayleigh random variable i given by: 3. AR Order: In the propoed recurive approach, a conventional tet could be performed on the reflection coefficient, at each order, to evaluate the quality of the current etimation baed on any uual local criteria [24], [26], [31], [40]. Nonethele, if we aume that all ignal in the tudied image are true AR of order lower than p, a global criteria may be derived from the following tatitical propertie: the unmoothed etimate ˆγ (p) ( ) 1/2 π SNR(ˆγ (k) )= 1.91,k >p. (32) 4 π In our experiment, SNR(ˆγ (p) )ifirtevaluatedand compared to thi value. If the limit i reached, then the true AR order i aumed to be k 1. A. Simulation Model V. Simulation Reult We propoe to tet our method on a et of imulated RF image. The imulated image are obtained by uing the ytem-baed approach introduced by Bamber and Dickinon [41] and later developed by everal author [18], [42] [44]. In it implet form, the RF image i obtained through the patial convolution between a tiue repone and a ytem kernel decribing the emitted acoutic pule. The tiue repone, when derived from a 2-D continuum approach [41], [42], may be written in the patial frequency domain by: T (u, v) = T (u, v) φ(u, v), (33) where u and v are the patial frequency variable. T (u, v) and φ(u, v), repectively, tand for the magnitude and phae of the tiue repone. The phae i aumed to be a uniform random ditribution; T (u, v) reflect the mean catter ize or pacing. For the continuum approach [43], a Gauian form may be ued leading to: T (u, v) = T 0 exp u 2 2 l 2 u exp v2 2 l 2 v, (34)

8 gorce et al.: radio frequency ultraound image and pectral feature etimation 1711 A RF line y j i built by performing a patial 2-D convolution between the ytem kernel and the tiue repone [18], [43] a follow: [ ] 2 y j (t) = t 2 h(x, y, t) x,y T (x, y), x=0;y=j. y (37) Fig. 3. Compoite tiue function T (x, y) correponding to the numerical phantom ued for the imulation tudy. Three media have been imulated (exterior, wall, interior) with correlation length equal to, repectively, 60 µm, 88 µm, and 10 µm. where T 0 i a parameter allowing to fix the cattering power. l u and l v are given by: l u = 1, 2πl x l v = 1 (35). 2πl y In thee equation, l x and l y tand for the correlation length of the medium reflecting the mean catterer ize (poibly aniotropic) of the tiue. Thi frequency approach allow to build homogeneou tiue repone image, by the ue of the invere Fourier tranform. Becaue in thi paper we deal with dicontinuitie, we adapt thi method in order to obtain image containing a patchwork of different kind of tiue. The piecewie homogeneou tiue repone i built in three tep. Firt, N homogeneou tiue image are built in the frequency domain, according to (33). Second, the invere 2-D Fourier tranform i applied independently to the N tiue image. Third, the o-obtained patial-domain tiue function are combined, yielding the final compoite patial tiue function T (x, y). Fig. 3 illutrate the o-obtained tiue repone for a combination of three different media. The ytem kernel i decribed, in a 2-D pace, a a Gauian-haped modulated inuoid. The propagation in direction x i obtained by introducing the uual propagation term x c 0 t,(c 0 being the ound peed): h(x, y, t) =h 0 exp (x c0t/2) 2 σ x 2 2 in y2 2 σ 2 y (2πf 0 ( t 2x c 0 )), (36) where h 0 tand for the pule amplitude and σ x, σ y for the pule width, repectively, along and perpendicular to the propagation axi. where j indicate the j th line i proceed and y i the lateral reolution. Thi model i uitable to introduce the effect of attenuation, focalization, and diffraction. Thi i an important apect to tet the ability of the method to recover not only dicontinuitie but alo the mooth pectral variation induced by uch effect. Thee effect modify the ytem kernel, with repect to the location in the image plane. Therefore, a patially variant ytem kernel i ued with patially varying parameter, h 0 (x, y), σ x (x, y), σ y (x, y), and f 0 (x, y). Attenuation i introduced by modifying the pule amplitude and the central frequency along the propagation path [22], [23]: f c (i, j) =f 0 i 2α k,j x σf 2, k=0 h 0 (i, j) =h 0 e (f 0 2 f2 c (i,j)) 2σ 2 f, (38) where i tand for the location in depth, x for the axial reolution, and α k,j for the attenuation coefficient of the tiue located at depth i in RF line j. Under a linear approximation, diffraction-focalization effect may be imulated uing a filtering approach [44] [46]. In our cae thi filtering i introduced uing a cloedform analytic expreion of h 0 (x, y), f c (x, y), and σ x (x, y), σ y (x, y). Thee variation are introduced under a polynomial form. The coefficient of the polynom are computed to fit the diffraction-focalization of the pule a meaured from the experimental imaging ytem (ee next ection) on an homogeneou phantom. B. Simulated Image The numeric phantom i built according to Fig. 3. Three part are defined the exterior, the wall, and the interior media having, repectively, a correlation lag equal to l c = 60 µm, 88 µm, 10 µm. The backcatter power around 4.5 MHz from the wall medium i choen a a power reference. The imulation parameter are et uch a the relative exterior and interior backcattered power are, repectively, equal to 2dBand 10 db. The ytem kernel at the focal point i defined with a central frequency f 0 of 4.5 MHz, a frequency bandwidth given by σ f =0.95 MHz, and a lateral width of 1.6 mm at 40 db. The RF image contain 128 line of 7.9 cm in depth (2048 ample), with a lateral pacing of 0.6 mm. The ampling frequency i et to 20 MHz. The correponding patial reolution, for a ound peed of 1500 m/, i equal to 32.5 µm. The attenuation coefficient for the three media in the imulation have been et to α out =0.1 Nep/cm,α wall =

9 1712 ieee tranaction on ultraonic, ferroelectric, and frequency control, vol. 49, no. 12, december 2002 Fig. 4. Envelope image of the numeric phantom with a tationary ytem kernel (a) and with a patially varying kernel (b) mimicking the effect of attenuation, diffraction, and focalization Nep/cm, α in =0.02 Nep/cm. The envelope image imulated by our approach for the tiue mimicking phantom with either a tationary (nor attenuation, nor diffraction) or a patially varying kernel are provided in Fig. 4. C. Spectral Etimation Spectral etimation i performed on the complex envelope of the RF ignal. The complex envelope i computed by uing the Hilbert tranform and a central (demodulation) frequency of 5 MHz. The reulting complex envelope i then downampled at 10 MHz. In order to provide the complex TS image, the oobtained complex envelope ignal are plit into independent, half-length, overlapping window of 16 ample. Thi correpond to a length of 1.2 mm, which i approximately equal to the RF pule duration. In term of dicontinuity detection, thi correpond to an accuracy of half of the 16 ample, i.e., 0.6 mm for both imulated and experimental data. The pectral etimation i performed with a 7th order AR model, with both 1t and 2nd order moothne contraint. The 2nd order moothing i required in order to preerve mooth variation due to attenuation and diffraction. In practice, at each AR order etimation, the olution correponding to the 1t order moothing i computed and i then ued a the initial olution for the 2nd order moothing. For the 1t order moothing, the hyperparameter are fixed according to Section IV-C,2. λ i choen two time higher than the limit fixed by (28) and (31), leading to λ (0) =2andλ (q) =0.75; q [1; p]. The theory developed in Section IV-C,2 i valid only for a 1t order moothing. In practice, our experiment have hown that a higher value i required for λ for accurate econd order moothing. Good reult were found by increaing λ by a factor larger than 2 and maller than 20. Thi i due to the fact that econd order moothing relie on difference between firt order derivative, wherea firt order moothing ue difference between value of the field. A the etimate of the derivative are more noiy, the weight of the econd order moothing ha to be more important [33], [36]. The behavior of the method i illutrated on the following pectral parameter: the mean pectral power, the central frequency of the pectra, and the whole pectra. The reult are compared to thoe obtained with a conventional uncontrained approach (i.e., tandard Burg algorithm). The theoretical value alo are provided in order to how the accuracy of the method in term of etimation and it efficiency in term of egmentation. D. Reult and Dicuion 1. Power Etimation: The power etimation i a conventional parameter ued for quantitative imaging a an approximation of integrated backcatter (IBS) [1], [20]. Fig. 5(a) how the theoretical power ditribution over the image plane. The meh repreentation i built with the power level a a function of the poition in the image plane. The power i repreented by the elevation together with the gray level. Thu, white area aociated to a high elevation correpond to high-power region; black area aociated to a low elevation correpond to low-power region. A log-cale i ued for the elevation, and the propagation direction i from left to right. It hould be noted that the power appear nontationary even in tiular homogeneou region, due to attenuation and diffraction. Fig. 5(b) how the power etimation from the imulated RF image, obtained with the tandard uncontrained Burg algorithm. Note that thi etimate i equivalent to the conventional power etimate baed on the zero-order correlation lag. The high variance of Burg etimate i characterized by an additive noie [31]. Note alo that the noie level i in the order of the level of dicontinuitie to be detected. The SNR, defined a the ratio between dicontinuity amplitude and the tandard deviation of additive noie, may be etimated from (27), and the level of dicontinuitie. Thi SNR i found for interior and exterior dicontinuitie to be SNR ext =1.3 and SNR int =6.5. Fig. 5(c) and (d) provide the reult obtained with the propoed method. In Fig. 5(c), only a 1t order moothing

10 gorce et al.: radio frequency ultraound image and pectral feature etimation 1713 For a ignal defined a a complex envelope of a Gauian pule, it may be hown [35] that the central frequency may be etimated from the reflection coefficient through: f c = Φ( γ(k) ) 2kπ f + f 0, (39) Fig. 5. Meh repreentation of the theoretical (a) and etimated (b), (c), (d) power over the whole RF image. The elevation a well a the gray level i proportional to the log-power of the RF ignal. The reult are obtained, repectively, by the Burg algorithm (b), and by the propoed method with a firt order moothing (c) and a econd order moothing (d). i applied; in Fig. 5(d) a 2nd order moothing i applied a decribed above. With the 1t order moothing, the method i able to detect the true dicontinuitie but alo generate fale dicontinuitie: 1t order moothing favor indeed piecewie contant olution, and, therefore, i not uitable to the etimation of the power mooth patial variation. On the oppoite, 2nd order moothing lead to much better reult becaue the low patial variation of the power are recovered and the dicontinuitie are properly preerved, which would allow an eay delineation of three region. A a conequence, only 2nd order moothed reult will be howninthefollowing. In order to allow a finer tudy of the accuracy of the method, Fig. 6 exhibit the power etimate along two RF line. In all plot, the model i drawn in gray, and the experimental etimate are provided in black. The firt RF line (number 20 over 128 line) croe only the exterior medium; the econd RF line (number 64 over 128 line) croe the three media. Fig. 6(a) and (b) provide the plot correponding to the Burg etimate. The additive noie i clearly een a random variation around the expected value. Fig. (c) and (d) plot the reult provided by our method. The effect of the appropriate moothing i een. The propoed method dratically reduce the etimation variance and yield unbiaed etimate. Moreover, all dicontinuitie are preerved. 2. Central Frequency Etimation: The central frequency i often ued in the field of tiue characterization, either a uch or for attenuation etimation [12], [13], [22], [23]. where f i the ampling frequency. In real image, the pule i not really Gauian and higher i the order k lower i the validity of (39). A good trade-off wa found with k = 2 in (39). Fig. 7(a) how the theoretical ditribution of the central frequency over the image plane. The meh i built with the central frequency a a function of the poition in the image plane. The central frequency i repreented by the elevation together with the gray level. Thu, white area aociated to a high elevation correpond to high central frequency region; black area aociated to a low elevation correpond to low central frequency region. The central frequency hift oberved at interface i due to the fact that the three media do not have imilar pectral propertie. Thu, the central frequency of the emitted pule (4.5 MHz) i hifted toward higher frequencie, depending on the pectral repone of tiue. Moreover, attenuation and diffraction introduce mooth variation of the central frequency. Attenuation progreively hift down the central frequency and the pule propagate in tiue-like media. Due to focalization, the central frequency i hifted down when the ditance from the focal point increae. In our imulation (and in real US image), thee effect are imultaneouly combined, leading to the frequency variation hown in Fig. 7(a). Fig. 7(b) how the central frequency etimation from the imulated RF image, obtained with the tandard uncontrained Burg algorithm. Note that the SNR i wore than the SNR obtained for power etimation. For the exterior and interior wall, the SNR may be approximated by, repectively, SNR ext < 1 and SNR int 1. The level of the noie i thu very high compared to the dicontinuity level. Fig. 7(c) provide the reult obtained with the propoed method, with 2nd order moothing. A for the power etimation, thee reult illutrate the ability of the method to reduce etimation variance and to recover dicontinuitie. Fig. 8 how the ame reult plotted for the middle line (64 over 128 line) croing the three media, and illutrate the accuracy of the etimation. A. Cardiac Data VI. Experimental Reult The real data have been provided by uing an experimental et-up developed at the Medical Imaging Computing laboratory of the Catholic Univerity of Leuven, Belgium [47]. Thi et-up i connected to the RF output of a commercial echocardiographic device (Tohiba SSH- 160A, Tokyo, Japan) with a phaed-array probe (5 MHz)

11 1714 ieee tranaction on ultraonic, ferroelectric, and frequency control, vol. 49, no. 12, december 2002 Fig. 6. The 1-D plot of the etimation of the local power along 2 RF line: 20th line on the left (a), (c), and 64th line on the right (b), (d). The depth i numbered in pixel, the power value are given in decibel. The reference (0 db) i arbitrary. In all plot, the theoretical model i drawn in gray; the experimental etimate are black. Frame (a) and (b) how the Burg etimate. Frame (c) and (d) how the etimate provided by our method. Fig. 7. Meh repreentation of the theoretical (a) and etimated (b), (c) central frequency over the whole RF image. The elevation a well a the gray level i proportional to the central frequency of the RF ignal. The reult are obtained, repectively, by the Burg algorithm (b), and by our method with a econd order moothing (c). Fig. 8. The 1-D plot of the etimation of the local central frequency along the 64th RF line. The depth i numbered in pixel, the frequencie are given in megahertz. In both plot, the theoretical model i drawn in gray; the experimental etimate are black. Frame (a) and (b), repectively, how the Burg etimate and the etimate provided by our method.

12 gorce et al.: radio frequency ultraound image and pectral feature etimation 1715 (depth ). Between thee region, dicontinuitie are detected and peak may be oberved (depth 42, 88, 106). Thee reult alo may be oberved on the 2-D ditribution given in Fig. 10(b). It may be noted from Fig. 10 that the peak appear only at location at which the heart wall i approximately perpendicular to the wave propagation direction. The peak can, therefore, be attributed to the pecular echoe correponding to tiue interface. Note that the above concluion would be difficult to draw from the etimate obtained without moothing and diplayed on Fig. 10(a) and (c). Fig. 9. Long axi view of the heart of a dog, obtained with an ultraound B-mode can uing a 5 MHz probe. and equipped with an analog RF interface. Thi ytem allow to digitize the complete ultraound ignal of at leat a complete heart cycle (50 image at a frame rate of 30 image per econd at a ampling frequency of 40 MHz). The image depth i equal to 15 cm. The propoed pectral etimation method wa applied to the RF image of a canine heart, acquired in long axi view during diatole. A conventional B-mode, ectorial recontruction image of the heart i given in Fig. 9. Note, however, that, in the following, the reult will be diplayed without ectorial recontruction, in order to analyze the behavior of the method without geometrical recontruction artefact. B. Reult and Dicuion The method wa applied under the ame condition a thoe ued for imulated image, except for the regularization parameter λ, which were et to lightly higher value: λ (0) =4andλ (q) =2;q [1; p]. Thi choice i done in agreement with expreion (28) and (31) developed in Section IV-C,2, to take into account the higher noie level preent in the experimental RF data. A for the imulation tudy, the reult hown were obtained uing 2nd order moothing. Fig. 10(a) and (b) how the 2-D ditribution of the power etimate provided by the Burg method Fig. 10(a) and the moothed etimate provided by the propoed method Fig. 10(b). In thee figure, wave propagation direction i from left to right. Fig. 10(c) diplay the profile of the power etimate obtained by thee two method along one RF line, located in the middle of the image. Depth along thi line i given in pixel unit (one pixel correponding to 0.6 mm). On Fig. 10(c), the moothed etimate allow to ditinguih two type of region: in region correponding to homogenou tiue, the power exhibit low variation due to attenuation and diffraction-focalization. Thee region correpond to the anterior wall (depth 22 38), the blood cavity (depth 43 85), and the poterior wall Fig. 11(a) how the 2-D ditribution of the moothed central frequency etimate provided by our method from the econd order reflection coefficient. The 2-D ditribution of the Burg etimate i not given a the very high variability of thee etimate make the meh repreentation uninterpretable. Fig. 11(b) how the profile of thi parameter along the middle RF line of the image, and allow to compare the reult provided by the Burg etimation and the propoed moothed etimation. Fig. 11(b) clearly how the high variance aociated with the Burg etimate, which doe not allow to differentiate tiue region. On the oppoite, the reult provided by the moothed etimation clearly dicriminate the blood cavity (depth 43 85) and the urrounding myocardium. The central frequency hift (about 0.2 Mhz) oberved between thee area can be attributed to the difference in backcatter repone of the aociated tiue, which thu weight in a different way the emitted pule pectrum. In the homogeneou region correponding to the cavity (depth 43 85) and the poterior wall region (depth ), an overall decreae of the central frequency may be oberved, which may be aociated to attenuation. Thi obervation doe not hold for the anterior wall, probably becaue attenuation and focalization effect compenate in thi region. The interface between thee homogeneou region are characterized by a downward hift (negative peak) of the central frequency. A noted for the power parameter, thi may be interpreted a being due to pecular reflection. Perfect geometric reflection can be indeed aociated to a frequency repone equal to one, but Rayleigh cattering may be aumed for the urrounding tiue, yielding a repone that varie a f n, n>1[5]. Fig. 12 how the full pectra etimated along the middle RF line of the cardiac image. The pectra were etimated uing a 4th order AR model. In the abence of an abolute reference model, a fully quantitative interpretation of thi reult i difficult. However, Fig. 12 illutrate the ability of the method to perform mooth pectral etimation while preerving pectral dicontinuitie. Smooth pectral variation are aociated with the three main homogeneou region (anterior and poterior wall, blood cavity), and the dicontinuitie are aociated to the interface between thee region, correponding to pecular reflection.

13 1716 ieee tranaction on ultraonic, ferroelectric, and frequency control, vol. 49, no. 12, december 2002 Fig. 10. Meh repreentation (a), (b) and line profile (c) of the local power etimate obtained by the Burg method (a) and by our method (b). The elevation a well a the gray level i proportional to the log-power of the RF ignal in (a) and (b). In (c) the 1-D plot repreent the etimation of the local power along the 64th RF line croing the heart cavity. Fig. 11. Meh repreentation (a) and line profile (b) of the local central frequency etimate obtained by the propoed method. In (a), the elevation a well a the gray level i proportional to the log-power of the RF ignal. In (b) the 1-D plot repreent the etimation of the local central frequency along the 64th RF line croing the heart cavity. VII. Concluion The primary goal of thi tudy wa to etablih a method for robut, hort-term pectral etimation of RF ultraound image, i.e., yielding low variance of the etimate and preerving pectral dicontinuitie. The propoed approach i baed on AR pectral etimation, contrained through nonquadratic penalty function. The method wa teted on a numerical imulation baed on a Gauian-haped ultraound pule propagating through three media having different acoutic propertie. Simulation reult how that the method provide unbiaed and low variance (mooth), hort-term etimate of pectral parameter uch a the power, the central frequency, or the whole pectrum. Moreover the pectral dicontinuitie between adjacent media are recovered. The main propertie of the method mooth etimate and dicontinuitie preervation were confirmed on the reult obtained from experimental canine cardiac RF image. In the abence of abolute reference, the etimated patial variation of the pectral parameter (power, central frequency, whole pectrum) were found conitent with the propagation general propertie: low decreae of the log-power and central frequency in homogeneou region located in the far field, with rapid jump/dicontinuitie at the interface between different tiue. The propoed method offer the perpective to create parametric image baed on reliable, pectrally de-

14 gorce et al.: radio frequency ultraound image and pectral feature etimation 1717 Fig. 12. A 3-D viualization of the time-frequency etimation performed along the 64th line croing the blood cavity and the myocardium wall. rived feature (uch a attenuation, integrated backcatter, backcatter coefficient, etc.) and it capacity to recover dicontinuitie provide the firt tep toward pectrally baed egmentation. Moreover, becaue thi approach only require that the pectral content to be recovered may be faithfully repreented through an AR model, we believe it could be profitably applied to a wide variety of ultraound image. Appendix A Lit of Symbol a (p) (k): the k th coefficient of the AR (p) model. AR (p) : autoregreive model (AR) oforderp. b c: line proce variable aociated with clique c. B: B = {b c; c C (n) }. c C (n) : a clique of degree n. c 0 : ound peed of the ultraound wave in human tiue. f 0 : central frequency of the emitted pule. f c(i, j): central frequency of the j th RF ignal at depth i. f : ampling frequency. h(i, j): time varying ytem kernel for imulating the j th RF line. l x,l y,l c: correlation length characterizing the tiue repone. N : Neighborhood ytem aociated with TS image. N : et of all cell neighboring. : indice of a cell = (i,j) containing a hort RF ignal. S: et of all cell in the TS image. P yy(f): power pectral denity (PSD) of y. U pot(x/y): poterior cot function of X under the knowledge of Y. U prior (X): prior cot function of X. U like (Y/X): likelihood of X under the knowledge of Y. V c(x): potential function aociated with the clique c. x (q) : vector of unknown at order q aociated with cell. X: X = {x ; S}. y : windowed RF ignal aociated with a cell. Y : Y = {y ; S}. δ: caling factor aociated with the Φ-function allowing to adapt the level of dicontinuitie. ɛ: coefficient aociated with the Φ-function allowing to adjut the accuracy of the moothing. ɛ(n): input noie ignal of the AR model. Φ(x): Φ-function. γ (q) : q th reflection coefficient aociated with cell. Γ (q) : Γ (q) = {γ (q) ; S}. λ: weighting factor allowing to adjut the relative influence of prior and likelihood cot function. σ 2 : input noie variance. σ (q)2 : input noie variance of the AR (q) model aociated with cell. ˆρ (q) : unregularized log-power etimate at order q aociated with cell. π (q) : log-power variable of the AR (q) model aociated with cell. Π (q) : Π (q) = {π (q) ; S}.. : conjugate of x. ˆ.: conventional unmoothed etimate of a variable or a vector..: etimate of a variable or a vector. Reference [1] B. Bijnen, M. C. Herregord, J. Nuyt, G. Vandeweghe, P. Sueten, and F. van de Werf, Acquiition and proceing of the radio-frequency ignal in echocardiography: A new global approach, Ultraound Med. Biol., vol. 20, no. 2, pp , [2] G. Davion, C. S. Hall, J. G. Miller, M. Scott, and S. A. Wickline, Ultraonic tiue characterization of end-tage dilated cardiomyopathy, Ultraound Med. Biol., vol. 21, no. 7, pp , [3] M. F. Santarelli and L. Landini, A model of ultraound backcatter for the aement of myocardial tiue tructure and architecture, IEEE Tran. Biomed. Eng., vol. 43, no. 9, pp , [4] A.F.W.vanderSteen,H.Rijterborgh,C.T.Lancée, F. Matik, R. Kram, P. D. Verdouw, J. R. T. C. Roelandt, and N. Bom, Influence of data proceing on cyclic variation of integrated backcatter and wall thickne in tunned porcine myocardium, Ultraound Med. Biol., vol. 23, no. 3, pp , [5] K. A. Wear, M. R. Milunki, S. A. Wickline, J. E. Perez, B. E. Sobel, and J. G. Miller, Differentiation between acutely ichemic myocardium and zone of completed infarction in dog on the ba-

15 1718 ieee tranaction on ultraonic, ferroelectric, and frequency control, vol. 49, no. 12, december 2002 i of frequency-dependent backcatter, J.Acout.Soc.Amer., vol. 85, no. 6, pp , [6] K. A. Wear, M. R. Milunki, S. A. Wickline, J. E. Perez, B. E. Sobel, and J. G. Miller, Contraction-related variation in frequency dependence of acoutic propertie of canine myocardium, J. Acout. Soc. Amer., vol. 86, no. 6, pp , [7] S. A. Wickline, E. D. Verdonk, B. E. Sobel, and J. G. Miller, Identification of human myocardial infarction in vitro baed on the frequency dependence of ultraonic backcatter, J. Acout. Soc. Amer., vol. 91, no. 5, pp , [8] S. A. Wickline, E. D. Verdonk, K. A. Wong, R. K. Shepard, and J. G. Miller, Structural remodeling of human myocardial tiue after infarction; quantification with ultraonic backcatter, Circulation, vol. 85, pp , [9] K. A. Wear, R. F. Wagner, M. F. Inana, and T. J. Hall, Application of autoregreive pectral analyi to ceptral etimation of mean catterer pacing, IEEE Tran. Ultraon., Ferroelect., Freq. Contr., vol. 40, no. 1, pp , [10] K. A. Wear, R. F. Wagner, and B. S. Garra, High reolution ultraonic backcatter coefficient etimation baed on autoregreive pectral etimation uing Burg algorithm, IEEE Tran. Med. Imag., vol. 13, no. 3, pp , [11] F. S. Cohen, G. Georgia, and E. J. Halpern, WOLD decompoition of the backcatter echo in ultraound image of oft tiue organ, IEEE Tran. Ultraon., Ferroelect., Freq. Contr., vol. 44, no. 2, pp , [12] S. L. Bridal, P. Fornè, P. Bruneval, and G. Berger, Parametric (integrated backcatter and attenuation) image contructed uing backcattered radio frequency ignal (25 56 MHz) from human aortae in vitro, Ultraound Med. Biol., vol. 23, no. 2, pp , [13] T. Spencer, M. Ramo, D. M. Salter, T. Anderon, P. P. Kearney, R. Sutherland, K. A. A. Fox, and W. N. McDicken, Characterization of atheroclerotic plaque by pectral analyi of intravacular ultraound: An in vitro methodology, Ultraound Med. Biol., vol. 23, no. 2, pp , [14]L.S.Wilon,M.L.Neale,H.E.Talhami,andM.Appleberg, Preliminary reult from attenuation-lope mapping of plaque uing intravacular ultraound, Ultraound Med. Biol., vol. 20, no. 6, pp , [15] P. Chaturvedi and M. F. Inana, Autoregreive pectral etimation in ultraonic catterer ize imaging, Ultraon. Imag., vol. 18, no. 2, pp , [16], Bayeian and leat quare approache to ultraonic catterer ize image formation, IEEE Tran. Ultraon., Ferroelect., Freq. Contr., vol. 44, no. 1, pp , [17] E. J. Boote, E. L. Maden, J. A. Zagzebki, and T. J. Hall, Intrument-independent acoutic backcatter coefficient imaging, Ultraon. Imag., vol. 10, pp , [18] M. F. Inana and T. J. Hall, Parametric ultraound imaging from backcatter coefficient meaurement: Image formation and interpretation, Ultraon. Imag., vol. 12, pp , [19] J. A. Zagzebki, X. L. Yao, E. J. Boote, and Z. F. Lu, Quantitative backcatter imaging, in Ultraonic Scattering in Biological Tiue. K. K. Shung and G. A. Thieme, Ed. Boca Raton, FL: CRC Pre, 1993, pp [20] A. D. Waggoner, J. E. Perez, J. G. Miller, and B. E. Sobel, Differentiation of normal and ichemic right ventricular myocardium with quantitative two-dimenional integrated backcatter imaging, Ultraound Med. Biol., vol. 18, no. 3, pp , [21] Z. F. Lu, J. A. Zagzebki, R. T. O Brien, and H. Steinberg, Ultraound attenuation and backcatter in the liver during prednione adminitration, Ultraound Med. Biol., vol. 23, no. 1, pp. 1 8, [22] T. Baldeweck, P. Laugier, A Herment, and G. Berger, Application of autoregreive pectral analyi for ultraound attenuation etimation: Interet in highly attenuating medium, IEEE Tran. Ultraon., Ferroelect., Freq. Contr., vol.42,no.1,pp , [23] J. M. Girault, F. Oant, A. Ouahabi, D. Kouamé, and F. Patat, Time-varying autoregreive pectral etimation for ultraound attenuation in tiue characterization, IEEE Tran. Ultraon., Ferroelect., Freq. Contr., vol. 45, no. 3, pp , [24] G. Kitagawa and W. Guerh, A moothne prior time-varying AR coefficient modeling of nontationary covariance time erie, IEEE Tran. Automat. Contr., vol. 30, no. 1, pp , [25] J. F. Giovannelli, G. Demoment, and A. Herment, A Bayeian method for long AR pectral etimation: A comparative tudy, IEEE Tran. Ultraon., Ferroelect., Freq. Contr., vol. 43, no. 2, pp , [26] J. F. Giovannelli, Etimation de caractéritique pectrale en temp court. Application à l imagerie Doppler, Ph.D. diertation, Univerité Pari-Sud, Oray, France, [27] J. M. Gorce, D. Friboulet, M. Robini, B. Bijnen, and I. E. Magnin, Regularized autoregreive model preerving patial dicontinuitie for analyi of radio-frequency (RF) ignal in echographic image, in 16ieme Colloque ur le Traitement du Signal et de Image (GRETSI), 1997, vol. 2, pp , Grenoble, France. [28] J. M. Gorce, D. Friboulet, J. D hooge, B. Bijnen, and I. E. Magnin, Regularized autoregreive model for a pectral etimation cheme dedicated to medical ultraonic radio-frequency image, in Proc. IEEE Ultraon. Symp., 1997, pp [29] F. Barbareco, Half-quadratic regularization of time-frequency AR analyi for recovery of abrupt pectral dicontinuitie and their detection by a recurive Siegel metric baed on information geometry, in EUSIPCO-98, 1998, pp [30] F. Barbareco, Extrinic and intrinic entropic prior for timefrequency AR analyi: Half-quadratic regularization and recurive iegel metric baed on information riemannian geometry, in PSIP-99, 1999, pp [31] S. M. Kay, Modern Spectral Etimation: Theory and application. er. Alan V. Oppenheim, Signal Proceing Serie, Englewood Cliff, NJ: Prentice Hall, [32] S. Geman and D. Geman, Stochatic relaxation, gibb ditribution, and Bayeian retoration of image, IEEE Tran. Pattern Anal. Machine Intell., vol. 6, no. 6, pp , [33] D. Geman and G. Reynold, Contrained retoration and the recovery of dicontinuitie, IEEE Tran. Pattern Anal. Machine Intell., vol. 14, no. 3, pp , [34] S. Z. Li, Markov Random Field Modeling in Computer Viion. er. Computer Science Workbench, Berlin: Springer-Verlag, [35] J. M. Gorce, Analye pectrale locale par modéliation autorégreive patialement régulariée. Application aux image de radiofréquence en échocardiographie ultraonore, Ph.D. diertation, INSA-Lyon, France, [36] A. Blake and A. Zierman, Viual Recontruction. Cambridge, MA: MIT Pre, [37] M. Baeville, Ditance meaure for ignal proceing and pattern recognition, Signal Proce., pp , [38] P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, Determinitic edge-preerving regularization in computed imaging, IEEE Tran. Image Proceing, vol. 6, no. 2, pp , [39] S. Geman and D. E. McClure, Bayeian image analyi: An application to ingle photon emiion tomography, in Proc. Statit. Comput. Sect., 1985, pp [40] G. Kitagawa and W. Guerh, A moothne prior long AR model method for pectral etimation, IEEE Tran. Automat. Contr., vol. 30, no. 1, pp , [41] J. C. Bamber and R. J. Dickinon, Ultraonic B-canning: A computer imulation, Phy. Med. Biol., vol. 25, no. 3, pp , [42] J. Meunier, Analye dynamique de texture d échocardiographie bidimenionnelle du myocarde, Ph.D. diertation, Québec, Canada: Univerité demontréal, [43] J. Meunier and M. Bertrand, Echographic image mean gray level change with tiue dynamic: A ytem-baed model tudy, IEEE Tran. Biomed. Eng., vol. 42, no. 4, pp , Apr [44] L. X. Yao, J. A. Zagzebki, and E. L. Maden, Backcatter coefficient meaurement uing a reference phantom to extract depth-dependent intrumentation factor, Ultraon. Imag., vol. 12, pp , [45] V. Roberjot, S. L. Bridal, P. Laugier, and G. Berger, Abolute backcatter coefficient over a wide range of frequencie in a tiue-mimicking phantom containing two population of catterer, IEEE Tran. Ultraon., Ferroelect., Freq. Contr., vol. 43, no. 5, pp , 1996.

16 gorce et al.: radio frequency ultraound image and pectral feature etimation 1719 [46] M. F. Inana, T. J. Hall, and L. T. Cook, Backcatter coefficient etimation uing array tranducer, IEEE Tran. Ultraon., Ferroelect., Freq. Contr., vol. 41, no. 5, pp , [47] B. Bijnen, Exploiting radiofrequency information in echocardiography: Acquiition, proceing and illutrated application, Ph.D. diertation, Katholieke Univeriteit te Leuven, Belgium, Jean-Marie Gorce wa born in 1970 in Belfort, France. In 1993 he received the engineering degree (electrical engineering) and in 1998 the Ph.D. degree (acoutic) both from the National Intitute for Applied Science of Lyon (INSA-Lyon, France). He worked from 1998 to 1999 at the Reearch Center of Bracco Reearch S.A. in Geneva, Switzerland. Jean-Marie Gorce i currently an aociate profeor at the Center for Innovation in Telecommunication and Integration of Service (CITI) of INSA-Lyon. Hi reearch interet include ignal modeling and propagation imulation applied to acoutic imaging a well a to radiocommunication. Deni Friboulet received the engineering degree (electrical engineering) in 1984 and in 1990 the Ph.D. degree (biomedical engineering), both from the National Intitute for Applied Science of Lyon (INSA-Lyon, France). From 1990 to 1991 he worked at the Intitute of Mathematic and Computer Science in Medicine at the Univerity of Hamburg (Germany) a a potdoc. Since 1991 he ha worked a an aociate profeor at the Reearch and Application Center for Image and Signal Proceing (CREATIS, Lyon, France). CREATIS i a reearch unit of the french National Center for Scientific Reearch (CNRS), common to INSA- Lyon and the Univerity Claude Bernard-Lyon. Hi reearch interet include ignal modelling, egmentation and motion etimation, applied to echographic ultraound image. Igor Dydenko wabornincracow,poland, in He received the engineering degree in mechanic and the mater degree in indutrial automation from the National Intitute of Applied Science (INSA) in Lyon, France, both in He i currently working toward the Ph.D. degree in medical image proceing at the Reearch and Application Center for Image and Signal Proceing (CREATIS, Lyon, France). CREATIS i a reearch unit of the french National Center for Scientific Reearch (CNRS), common to INSA-Lyon and the Univerity Claude Bernard-Lyon. Hi reearch interet include modeling and ignal proceing of ultraonic cattering a well a RF image egmentation and characterization. Jan D hooge (M 99) wa born in Sint- Niklaa, Belgium, in He received the M.Sc. and Ph.D. degree in phyic at the Catholic Univerity of Leuven, Belgium, in 1994 and 1999, repectively. Hi diertation tudied the interaction of ultraonic wave and biological tiue by mean of computer imulation. Dr. D hooge i currently a reearch fellow at the Medical Imaging Computing (MIC) laboratory of the Department of Electrical Engineering of the Catholic Univerity of Leuven and work in cloe collaboration with the Department of Cardiology and Acoutic of the ame univerity. He i a member of the Acoutical Society of America and IEEE. In 1999 he won the young invetigator award of the Belgian Society of Echocardiography, and in 2000 he wa nominated for the young invetigator award of the European Society of Echocardiography. Hi current reearch interet include myocardial tiue characterization and train imaging. Bart H. Bijnen (S 89 M 90) wa born in Bilzen, Belgium, in He received a Mater degree in electronic engineering in 1990 and a Ph.D. degree in medical cience in 1997 from the Catholic Univerity of Leuven, Belgium. From 1990 to 1998 he wa a reearcher at the Department of Cardiology of the Univerity Hopital Gathuiberg and the Univerity of Leuven. Since 1998 he ha been an aociate profeor at the Faculty of Medicine and coordinator of teaching coure in cardiac imaging. Hi main reearch topic i cardiac imaging, with a pecial focu on echocardiography. Current project involve myocardial tiue characterization baed on RF data, myocardial deformation analyi (train- and train-rate imaging) baed on ultraound and contrat echocardiography. Iabelle E. Magnin (M 85) received the received the engineering degree from Ecole Catholique d Art et Mtier (ECAM, Lyon, France) in 1977 and the Doctorat d état è Science in 1987 from the National Intitute for Applied Science (INSA-Lyon, France). She i a reearcher from the French National Intitute for Health and Medical Reearch (INSERM). She i the co-director of the Reearch and Application Center for Image and Signal Proceing (CREATIS, Lyon, France). CREATIS i a reearch unit of the french National Center for Scientific Reearch (CNRS), common to INSA-Lyon and the Univerity Claude Bernard-Lyon. Her main interet concern modeling, medical image proceing, dynamic imaging, and invere problem. She i involved in more than 100 publication. She i a member of the IEEE.