Optimal Apodization Design for Medical Ultrasound using Constrained Least Squares. Part I: Theory D. A. Guenther and W. F. Walker

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1 Optimal Apodization Deign for Medical Ultraound uing Contrained Leat Square. Part I: Theory D. A. Guenther and W. F. Walker Abtract Aperture weighting function are critical deign parameter in the development of ultraound ytem becaue beam characteritic affect the contrat and point reolution of the final output image. In previou work by our group, we developed a metric which quantifie a broadband imaging ytem contrat reolution performance [1]. We now utilize thi metric to formulate a novel general ultraound beamformer deign method. In our algorithm we ue contrained leat quare (CLS) technique and a linear algebra formulation to decribe the ytem point pread function (pf) a a function of the aperture weighting. In one approach we minimize the energy of the pf outide a certain boundary and impoe a linear contraint on the aperture weight. In a econd approach we minimize the energy of the pf outide a certain boundary while impoing a quadratic contraint on the energy of the pf inide the boundary. We preent detailed analyi for an arbitrary ultraound imaging ytem and dicu everal poible application of the CLS technique, uch a deigning aperture weighting to optimize cytic reolution and improve the ytem depth of field. Simulation reult are preented in an accompanying paper [2]. Introduction The determination of array aperture weight which produce a yntheized beam pattern with a narrow mainlobe and low idelobe i a claical problem with a rich hitory in the ignal proceing literature. Dolph ued Chebyhev polynomial to calculate aperture weight for a uniformly paced, continuou wave linear array that achieved the minimum poible beamwidth 1

2 for a given maximum idelobe level [3]. Taylor expanded thi formulation to achieve tapered idelobe further away from the mainlobe for continuou aperture [4], and Villeneuve applied it to dicrete array [5]. Alfred H. Nuttall, in hi paper [6], improved upon the Blackman-Harri window to achieve beam pattern whoe maximum idelobe are minimized. Wherea thee previou paper focued on uniformly paced array, Olen and Compton developed an iterative procedure uing an arbitrarily haped adaptive array to produce the deired idelobe behavior by uing a recurive feedback procedure [7]. Teng and Griffith alo produced a imple iterative algorithm that can be ued to find array weight for nonuniform geometrie to produce beam pattern with a given look direction and minimum energy in the idelobe [8]. An intereting outcome of their method allowed for the deign of beam pattern where the deired idelobe repone could vary with angle. Although thee previou method produced excellent reult narrowband aumption, computational complexitie, and iterative procedure limit their applicability to general ultraound beamformer deign. Coniderable gain can be made in computation time with the ue of leat quare method. In fact, over the lat two decade many author have developed contrained leat quare algorithm for the deign of FIR filter [9-12]. Thee method typically minimize the error of the filter over a certain frequency band with repect to ome deired filter repone. For example, Selenick decribed a contrained leat quare approach to deign FIR filter that did not require the pecification of a tranition band of frequencie between the paband and topband. By etting up a minimization problem on the l 2 error of the filter amplitude repone ubject to linear equality contraint, Selenick derived filter with minimum error and devoid of Gibb phenomenon [9]. Other author ued leat quare method to produce eigenfilter, or filter that minimize a quadratic error meaure in the paband and 2

3 topband [10-12]. Later in a erie of paper, Er et al. employed a variety of contrained leat quare technique to yntheize arbitrary array pattern ubject to different criteria uch a idelobe level and mean quared idelobe energy. Their algorithm employ linear and quadratic contraint to achieve array pattern for general array geometrie which produce beam pattern that are highly directional with very low idelobe [13-18]. The rich hitory of array pattern ynthei optimization ha been only recently applied to medical ultraound imaging and mot application have been pecialized [19-23]. For example, Ebbini and Cain [19] propoed a method for yntheizing multiple focal region on ingle tranmit event for application in hyperthermia treatment via ultraound. Li et al. [20] ued a total leat quare method to compenate for pf degradation due to dead array element or element blocked by acoutically opaque window in the interrogated media. Recently, Wilkening et al. deigned optimal FIR filter for improved image contrat in contrat agent imaging [22] and FIR filter that increaed the depth of field for dynamic receive focuing [23]. Although inightful and ueful, thee method failed to addre the larger problem of general beam pattern ynthei given arbitrary array. Previouly, our group developed a general aperture deign tool, upported by rigorou theory that i applied to the deign of aperture weighting function for arbitrary ytem deign [24], [25]. The method utilized a minimum um quared error (MSSE) formulation between the ytem pf and the deired or goal pf. One trength of the approach i that it allowed for full beam optimization given ytem parameter obtained through theory, imulation, or experiment. The method i ueful becaue it allow for the deign of any controllable ytem parameter in a traightforward, rigorou, time efficient manner. Ranganathan approach, although extremely ueful in aiding the deign of prototype ytem, uffer from the lack of a quantitative meaure detailing how ytem performance 3

4 change with repect to a deviation in ytem parameter. Furthermore, the approach offer no guidance in the election of an appropriate goal point pread function. Thee hortcoming make ytem optimization uing the MSSE approach difficult. The method of apodization profile deign preented in thi paper i general enough to be applied to any coherent imaging ytem and i imilar to many of the previou array pattern ynthei technique utilizing contrained leat quare (CLS). For example, our linearly contrained leat quare (LCLS) formulation i imilar to the array pattern ynthei technique by Teng [8], and our quadratically contrained leat quare (QCLS) formulation i imilar to the contrained eigenfilter deign [10], [26]. However while thee prior analye were for a ingle carrier frequency, we ue a broadband formulation. Keitmann-Curde et al. [27] previouly developed an algorithm imilar to our QCLS formulation which generated apodization profile for ultraound imaging with minimum idelobe energy of the two dimenional pace-time pf. Recently Schwann et al. ued two different reolution criteria to deign optimal frequency dependent apodization profile [28], a method whoe goal are imilar to our of improving image contrat. However, their multiple objective formulation require computationally expenive iterative method to arrive at one Pareto optimum olution, or a olution where further improving one objective necearily degrade all other [29]. Further review of the difference between our LCLS and QCLS method and the technique mentioned above will be dicued in more detail later in thi manucript. Since the ultraound ytem beam characteritic fundamentally affect the quality of the image, a great deal of effort i put into optimizing ytem parameter. Etimating the imaging performance of ultraound ytem i critical, both to characterize the fundamental imaging limit of the ytem, and to optimize image quality. It i poible to etimate the performance of 4

5 exiting ytem by imaging phantom or human ubject, but it i neceary during ytem deign to be able to determine imaging performance prior to ytem contruction. The ability to accurately predict performance enable ytem optimization, quantitative conideration of engineering tradeoff, and ignificantly reduce the time and cot invetment in ytem development. Synthei of beampattern in diagnotic ultraound receive a great deal of attention during ytem deign. The ytem patial impule repone, or point pread function, characteritic will determine uch parameter a point reolution and contrat in the reulting image. Thu control over mainlobe width and idelobe level i ignificant. Thee beam parameter are influenced by the ize of the active aperture, the frequency of the ultraound pule, the magnitude and phae (or time delay) of the weighting applied to the active element, and the pule length. Becaue o many factor affect the characteritic of the pf and there i no global parameter decribing pf quality, beamforming parameter are uually determined through iterative imulation and experimentation. A quantitative reolution metric i eential to guide optimization of ytem parameter, including the ytem pf. The mot common meaure of canner performance i the beamplot [30], which ha been adapted from RADAR. The -6dB beamwidth of the beamplot, the full width at half maximum (FWHM), and the beamwidth at other level are ued to etimate canner reolution. Sidelobe and grating lobe level are ued to etimate eventual image contrat. Although widely ued in medical ultraound, there are cenario in which the FWHM criterion indicate excellent performance, but actual image of tiue do not reveal important detail. Vilkomeron et al. addreed the limitation of the beamplot and propoed the concept of cytic reolution [31] in which performance wa quantified a the ize of a void that produced 5

6 a given contrat. The analyi, while novel and ueful, wa limited to narrowband circular aperture and neglected the axial dimenion. Johnon [32] further developed the contrat reolution metric to include a 3D broadband model for circular aperture and compared different imaging parameter uing maximum output contrat curve veru cyt diameter. Ütüner et al. [33] extended cytic reolution to a 3D broadband model for arbitrary aperture, but did not decribe it theoretical foundation, reulting in a limited undertanding of the formulation and it utility and drawback. A general cytic reolution metric wa previouly derived by our group [1]. Thi metric account for the effect of electronic noie and, under certain aumption, reduce to that decribed in [33]. Wherea, the FWHM criterion ometime provide mileading information about reolution in ultraound ytem, the cytic reolution metric identifie pecific point in the pf of the ytem that can be optimized to increae image quality and performance. Thi paper utilize the cytic reolution metric to guide optimization of apodization profile for coherent imaging ytem. Specifically, we deign optimal receive apodization profile for a 1D linear array; however, our theory can be applied to a 2D array of arbitrary geometry and can be ued to deign one way or two way apodization profile. We propoe two different method for optimal apodization deign. The firt algorithm minimize the energy of the pf outide ome pecified region ubject to a linear contraint of the apodization weight. We call the reultant weight the linearly contrained leat quare (LCLS) apodization profile. The econd algorithm minimize the energy of the pf outide ome region ubject to a quadratic energy contraint of the pf inide the boundary. We call the reultant weight the quadractically contrained leat quare (QCLS) apodization profile. 6

7 Our CLS apodization deign method return real weight and achieve a patial impule repone with minimum idelobe level in a leat quare ene given a pecified mainlobe area. We formulate the problem tarting from baic principle of acoutic wave diffraction theory and apply linear algebra technique to repreent the ytem pf. We generate a leat quare problem ubject to either a linear or quadratic contraint in order to minimize the energy outide a given mainlobe area in the pf. The algorithm can be applied to enhance the depth of field (DOF) in an imaging ytem a well a improve leion detectability in inhomogeneou cattering media. The algorithm i arguably optimal for detecting anechoic cyt via ultraound; however we believe it will alo improve ultraound ytem performance in general imaging application. Thi paper outline the theoretical decription of the contrained leat quare technique for deigning apodization profile for broadband, coherent imaging ytem, decribe a technique for reduced computational cot, and finally dicue example of application. Reult from imulation are preented in an accompanying paper [2]. Theory We preent two-way broadband formulation for the LCLS and QCLS apodization deign technique. The one-way broadband formulation can be expreed in a imilar manner; however we note that in mot ultraonic imaging application, apodization i typically applied only on receive and the two-way impule repone i of greater interet. Linear Algebra Formulation of the Broadband Spatial Impule Repone (pf) The acoutic preure field emanating from a tranducer during pule echo propagation at a ingle point in pace at a ingle intant in time can be expreed a the product of a propagation 7

8 matrix, S, and a et of aperture weighting, w. The propagation matrix ue uperpoition to decribe the contribution of each tranducer element at each field point at an intant in time. The propagation function may be derived from the Rayleigh-Sommerfeld diffraction equation derived in ([34, pp.46-50]) and may alo include a term relating to limited element angular repone [35]. Alternatively, the propagation matrix may be computed via broadband imulation or etimated experimentally. For our formulation, S i a function of the tranmit aperture weight, the excitation pule, and the individual element impule repone of the tranmit and receive aperture [24]. The two way pule echo propagation matrix, S, for a fixed tranmit aperture and a n element receive aperture at a total number of p point in three dimenional pace i: S = M p 1,1 2,1,1 1,2. M. L L O L 1, n. M p, n, (1) where i,j i the contribution of the jth element at the ith point in pace. The receive aperture weighting function, w, for each of the n element ued on receive can be written in vector form a: w = [ w w w L ] T, (2) w n where T denote the vector tranpoe operation. Uing (1) and (2), we can now write the complete two-way pule echo ytem pf, P, a follow: P = Sw, (3) 8

9 the propagation matrix multiplied by the receive weighting vector. Note that thi reult in the one dimenional column vector, P, of length p the total number of point in three dimenional pace where the ytem pf i meaured. Thi formulation can be expanded to decribe the pf a a function of time. In thi cae, the receive weighting would be a function of element number and time, eentially forming the coefficient of a FIR filter on each receive channel. Adequate patial and temporal ampling of the three dimenional pf yield huge propagation matrice, and therefore for thi and the accompanying paper we have limited our analyi to a ingle intant in time and two patial dimenion, azimuth and range. Clearly the elevation dimenion matter in planar ultraonic B- mode image, even with acoutic lene on linear array. However, retricting our analyi to two dimenion eae viualization of the algorithm while till providing meaningful reult. Cytic Reolution Metric The goal of the cytic reolution metric i to quantify the contrat reolution of an arbitrary broadband ultraound ytem. We refer the reader to [1] for a more detailed dicuion of the derivation of the metric and highlight the meaningful reult here. The metric completely characterize the 4D patiotemporal contrat performance for a ytem imaging an anechoic void. However, analyi at the intant in time when the received ignal i minimum (i.e. when a much of the pf energy a poible lie within the cyt) i uually ufficient. At thi ingle intant in time, the pf can be expreed a a function of 3D pace at the time of interet. The SNR i alo conidered at the time of interet ( SNR ). The contrat of the cyt relative to the background t o i defined a the ratio of the rm ignal received from the cyt to the rm ignal received from the background [1]: 9

10 C t0 2 Eout 1 + SNRto E, (4) = 1 + SNR tot 2 to where Eout i the pf energy outide the cyt and E tot i the total pf energy, both at time t o. Equation (4) decribe the contrat of an anechoic cyt, whoe ize and location are decribed by a mak, relative to background peckle obtained by an imaging ytem with a given pf and electronic SNR defined tatitically by SNR t o. Neglecting electronic noie, SNRt o become infinite and (4) can be modified to the equation for contrat preented in [33], which i imply the quare root of the ratio of the pf energy outide the cyt and the total pf energy: E out Ct o =. (5) Etot The contrat for cyt of different ize can be computed uing the above expreion for cytic contrat, and ytem performance can be characterized a a function of cyt ize a in [1, 31-33]. Thi metric can be ued for 4D patiotemporal analyi of broadband ultraound ytem, but 3D patial analyi uing (4) or (5) i typically adequate to characterize canner performance a temporal analyi doe not uually provide critical information. Note that in certain cae, it i valuable to compute the metric with cyt at different location to quantify the depth of field, the effect of dynamic focuing, and other factor pertaining to the hift variance of the imaging ytem. Note alo that while the metric can be ued to determine cytic reolution, it can alo be ued to optimize ytem parameter by computing contrat a a function of cyt ize and determining parameter value that maximize the contrat at the cyt ize of interet. Linearly Contrained Leat Square (LCLS) Apodization Deign 10

11 One conpicuou reult of the above reolution metric i that cytic contrat, or our objective function, i defined in term of the patiotemporal pf energy. In fact, contrat would be maximum if all the energy of the pf lay inide the void of the cyt. Wherea beamplot detail can be mileading about overall image quality, thi metric conider the pf globally to determine the impact on cytic contrat. Contrat improve when the pf energy outide the cyt boundary i reduced or the pf energy inide the cyt i increaed. The cytic reolution metric define a imple objective function for maximizing cytic contrat. The reader hould note that different objective function could be formulated, uch a improving point contrat, but for the dicuion preented here we focu on improving cytic contrat. Cytic contrat i degraded by the preence of pf energy outide the cyt. We minimize thi energy by olving for the et of receive aperture weight that, when applied to the ynthetic receive element repone, will yield a pf with minimum energy outide the deignated cyt boundary. Combining the above reolution metric with our linear algebra formulation of the pf, our beam ynthei problem become determining the vector of weight that minimize the pf energy outide the cyt boundary ubject to a linear contraint to avoid the trivial cae of all the receive weight et equal to zero. Thi i analogou to the problem of olving for the et of FIR filter coefficient that minimize the energy in the topband. Auming the pf i focued at the center of the cyt, the algorithm i initialized by electing the patial point of the pf which lie outide the cyt boundary. We form the aociated propagation matrix, S, which ha a many row a the number of point outide the cyt region and a many column a element in the active receive aperture. Therefore, each column of the S matrix i one focued ynthetic receive element repone at all the patial point outide the cyt boundary. Ranganathan MSSE beamformer deign approach outlined in [24] 11

12 and [25] ue focued or unfocued aperture propagation matrice. Although imaging cenario exit with unfocued aperture, for the CLS apodization deign technique preented here, we prefocu our two way pf o that the peak of the mainlobe lie in the center of the cyt boundary. Thi allow u to maximize the cot function in (5), cytic contrat. Uing unfocued aperture i poible with the CLS formulation, but unfocued aperture would not typically be ued for imaging cyt o a new cot function hould be derived to reflect the ytem intended application. The CLS algorithm calculate the weight which minimize the energy in the idelobe region while imultaneouly maintaining a peak gain at the center of the cyt. Thee weight are determined from the contrained leat quare problem: min Sw w 2 ubject to the linear contraint C T w = 1. (6) In thi expreion 2 denote the quare of the 2 l -norm and the row vector T C ha element correponding to the amplitude of each ynthetic receive element repone. More pecifically, T C i a vector of the amplitude of the receive element repone at the focu in the center of the cyt. The expreion in (6) i common in the ignal proceing literature and drawing upon [36] the optimal receive aperture weighting are given by: w T 1 T T 1 1 opt = ( S S ) C[ C ( S S ) C ], (7) where ( ) 1 denote the matrix invere operation. Equation (7) provide a imple method to calculate the receive weighting that will minimize the energy in the pf outide a pecified mainlobe region while imultaneouly achieving peak gain inide the mainlobe region. The optimal receive weight minimize Eout in (5) above, o we expect to ee improved cytic contrat 12

13 uing the LCLS apodization window over commonly ued window uch a the flat, Hamming and Nuttall [6] window. Linearly Contrained Leat Square Apodization Deign with Weighting Function In certain application, the pf characteritic at pecific patial poition may be more important than other becaue of the effect that the pf ha on point reolution and ytem contrat. For example, in hyperthermia application the ultraonic field pattern require high power level at ome point while reducing the power depoition at other potential hot pot [19]. In other application, it may be more important to reduce idelobe level than to preciely control the mainlobe. In thee cae and other we can incorporate a weighting function, g, that emphaize or deemphaize certain patial point in the pf during the apodization deign procedure. The LCLS apodization deign problem can be rewritten with the weighting function a: min w Sw g d 2 ubject to the linear contraint C T w = 1, (8) where gd i a diagonal p x p matrix with element of g along the 0 th diagonal. The element of g have a large value where minimizing pf energy i important and maller value where the pf energy i le critical. The olution for optimal receive weighting, drawing upon [36] i: w T T 1 T T T 1 1 opt = ( S gd gd S ) C[ C ( S gd gd S ) C ]. (9) Quadratically Contrained Leat Square (QCLS) Apodization Deign The LCLS apodization deign algorithm minimized the energy in the pf outide a given region of the mainlobe. By doing o the cytic contrat hould improve according to (5). Thi 13

14 analyi minimize the numerator of (5); however, it ignore the denominator. A a reult, although the energy outide the cyt will be minimized the total pf energy could alo be decreaed thu limiting cytic contrat improvement. We therefore develop an alternate approach where we minimize the energy of the pf outide a given boundary while at the ame time keeping the energy of the pf inide the boundary contant. Thi formulation become imilar to the earlier beam ynthei problem of creating eigenfilter [10], [26]. In the FIR eigenfilter deign cae, the energy contraint on the filter coefficient i uually jut the quadratic contraint w T w = 1 which contrain the total energy of the filter frequency repone to be unity. However, for broadband beamformer, thi contraint i meaningle and we mut devie a new formulation [26]. The modified quadratic contraint i traightforward given the cytic reolution metric. We wih to minimize Eout while maximizing Etot in (5), and we can change Etot to jut the pf energy inide the cyt boundary, E in. Note that thi problem can be et up a a multiple objective optimization problem [29], [37]. However, that approach doe not yield an intuitive optimal olution like utilizing the cytic reolution metric doe. Therefore, we chooe to formulate thi problem uing a quadratically contrained leat quare formulation: min w w S out 2 ubject to the quadratic contraint S in w = 1, (10) 2 where 2 denote the quare of the 2 l -norm, Sout i the propagation matrix for all the patial point of the pf lying outide the cyt boundary, and S in i the propagation matrix for all the patial point of the pf lying inide the cyt boundary. Note that the quadratic contraint eentially keep the energy of the pf inide the cyt contant. Drawing upon [26], [38-40] the optimal receive aperture weighting atifying the quadratic contraint i the generalized eigenvector, w eig, correponding to the minimum generalized eigenvalue reulting from the 14

15 generalized eigenvalue decompoition problem of S T S and T S out out in in S. The generalized eigenvalue problem [38, pp ] for a matrix pair, (A, B) both n x n matrice, i finding the eigenvalue, k, and the eigenvector, x k 0, uch that: Ax k = Bx. (11) k k The number of eigenvalue, k, i dependent upon the rank of matrix B. One of the main advantage of the QCLS technique i that no matrix inverion i required to olve for the optimal apodization profile, unlike the LCLS apodization deign. Quadratically Contrained Leat Square Apodization Deign with Weighting Function A with the LCLS apodization deign, a weighting function, g, can be added that emphaize or deemphaize certain region of the pf during the minimization proce. Rewriting the QCLS apodization deign problem above with the added weighting function we arrive at: min w g out S out w 2 ubject to the quadratic contraint gin S inw = 1, (12) 2 where gout i a diagonal p x p matrix with element of g aociated with the patial point of the pf outide the cyt boundary along the 0 th diagonal, and g in i a diagonal q x q matrix with element of g aociated with the patial point of the pf inide the cyt boundary along the 0 th diagonal. We olve thi problem by forming the Lagrangian and utilizing the neceary condition for a minimum: 15

16 T T T T T T L ( w, ) = w S g g S w ( w S g g S w 1), (13) out out out out in in in in where i the aociated Lagrange multiplier. Following the Kuhn-Tucker condition, a L neceary condition for a minimum i that = 0 w [37]. Thu taking the aociated partial derivative of the Lagrangian with repect to the weighting vector we arrive at: S T out g T out g out S out w T T = S g g S w. (14) in in in in Therefore the et of optimal receive weighting atifying (14) i again the generalized eigenvector, w eig, correponding to the minimum generalized eigenvalue reulting from the generalized eigenvalue decompoition problem of T T T T SoutgoutgoutSout and in ginginsin S. Reduced Computational Cot Through Symmetry Relation The computation of CLS apodization profile require ignificant reource due to the large propagation matrice and matrix invere operation (for the LCLS deign method). In order to reduce the computational complexity of the algorithm we take advantage of the lateral ymmetry preent in the ytem pf for ymmetric, non-teered aperture. Thi ymmetry mean that we can ue jut half of the ytem pf for the calculation of the optimal weighting. The propagation matrix, S, then become: S 1,1 = 2,1 M p / 2,1 1,2. M. L L O L 1, n. M p / 2, n, (15) 16

17 where S i a (p/2) x (n) matrix, coniting of the preure field at only p/2 point in pace for each element [1, 2, n]. The ymmetry of the non-teered receive aperture i another property that can be exploited to reduce computational cot. A hown in Fig. 1, pair of element can be grouped together that are the ame ditance from the center axi of the array. Pairing i poible becaue thee element hould have the ame weight applied, auming no beam teering and an even number of element in the aperture. Therefore the propagation matrix can be rewritten: S = M p / 1,1, n 2,1, n 2,1, n 1,2, n1. M. L L O L 1, n / 2, n / M p / 2, n / 2, n / 2+ 1, (16) where i,j,k i the repone at the point i in pace for the element j plu the repone for element k at the ame point in pace. The aperture weight mut alo be rehaped a: [ w w w w ] T w 1, n 2, n1 3, n2 n / 2, n / 2+ 1 = L, (17) where w i,j i the weight applied for element i and element j, repectively. The derivation i now analogou to that in (6) and (10) and the optimal weight can be determined directly. The ue of ymmetry reduce the ize of the propagation matrix, S, by a factor of 2 in each dimenion, for a total reduction of a factor of 4 in memory requirement. The computational aving will be even greater ince the neceary olution algorithm have polynomial cot a a function of matrix ize. 17

18 Fig. 1. Exploitation of ymmetry for reduced computational cot. The aperture i ymmetric about the center axi; therefore pair of element that would have the ame weight are grouped together. Alo, the lateral ymmetry in the pf about the ame center axi allow for analyzing jut one half of the pf. Thi ymmetry aume a ymmetric non-teered aperture. Application Although the CLS apodization profile dicued above were contructed to optimize cytic reolution, the technique are general enough to be applied in wide-ranging cenario. A few poible application are decribed. A. Improved Cytic Contrat and Improved Point Reolution The cytic reolution metric decribed in [1] tated that the point contrat of an ultraound ytem imaging a cyt i a function of the pf energy. Neglecting ytem SNR, we note that according to (5) we can improve the contrat of a cyt in two way. If we minimize the energy of the pf outide the cyt boundary, the numerator in (5) decreae and contrat improve. Furthermore, if we minimize the energy of the pf outide the cyt boundary and increae the 18

19 energy of the pf inide the cyt boundary, contrat will improve even more dramatically. The LCLS and QCLS method decribed above minimize the pf energy outide a pecified boundary ubject to a linear contraint on the weight or a quadratic contraint on the weight, repectively. Thu cytic contrat will be improved by uing optimal apodization profile. The LCLS apodization profile produce pf with narrow mainlobe and minimum idelobe energy. Thee profile eem to break the governing rule of windowing in ignal proceing: in order to achieve lower idelobe the mainlobe mut broaden, a reult exhibited by traditional apodization function. Therefore, the LCLS pf are more attractive for point imaging in general ultraound application, not only imaging anechoic leion. Although the LCLS deign approach improve cytic reolution, it i not truly optimal for imaging diffue leion and low echogenicity cyt. In thee cae a broader mainlobe may be deirable. Such profile can be deigned uing the QCLS approach. B. Enhanced Depth of Field The depth of field (DOF) of an ultraound imaging ytem i generally defined a the axial region over which the ytem i in focu, or the axial region over which the ytem repone remain imilar to the pf at the focu. Current method to improve DOF include tranmitting at a high f/#, dynamically receiving at low f/#, and dynamic receive apodization [41]. The implementation of thee technique lack formal theory decribing effectivene in improving DOF. Application of the CLS algorithm at every range yield receive weighting that force the pf at each interrogated range to have a pecific mainlobe width and the lowet poible idelobe energy outide that mainlobe. Applying dynamic receive apodization with thee weighting will produce imilar pf in range and improve the DOF. 19

20 C. Optimal Apodization for Harmonic Imaging The linear algebra formulation of the pf require linear uperpoition on receive but place no linearity contraint on tranmit. Conventionally, ultraound imaging ytem aume that the propagation of the ound pule on tranmit i linear and that the receive ignal ha the ame frequency content a that of the tranmitted pule. However, the propagation proce i ubtantially nonlinear and it i poible to receive echoe whoe energy content i hifted to harmonic of the fundamental tranmit frequency. Imaging with thee higher harmonic echoe can improve contrat and reolution in the reulting image. Our CLS technique can be adapted to calculate receive apodization profile that take nonlinear propagation into account. The nonlinear propagation of the tranmit beam can be determined analytically, experimentally, or through imulation and ubtituted into the linear algebra formulation of the pf. Auming linear propagation on receive, the algorithm will deign receive aperture function that minimize the energy of the two-way pf outide a pecified boundary. Equation 7 which i rewritten below, decribe the relationhip between the harmonic imaging cenario and the receive aperture weighting which will minimize the idelobe energy of the pf. Note that the propagation matrix S will have to take into account the nonlinear propagation effect of the tranmit acoutic beam: w T 1 T T 1 1 opt = ( S S ) C[ C ( S S ) C ]. D. Arbitrary pf hape for general imaging, hyperthermia or Doppler application In ome cenario it may be more important to achieve pf with greater idelobe rolloff. The weighted CLS formulation decribed above can achieve uch ytem repone by incorporating 20

21 a weight function that increae with ditance from the mainlobe. It i alo poible with the weighted CLS algorithm to deign pf with localized area of reduced energy. It hould be noted that we focued on producing apodization that minimized the pf energy in the region lying outide a (typically pherical) void. Thi procedure wa implemented in order to optimize cytic reolution. However, there i no need that the mainlobe and idelobe region be delineated according to the hape of a cyt. The CLS formulation can be adapted to deign optimal pf for a variety of ultraound application. In ultraound hyperthermia procedure, where control of the acoutic energy delivered to the tiue i of great concern [19], deigning pf with multiple mainlobe or hot pot while minimizing energy tranfer at other location could improve treatment efficacy a well a horten treatment time. It i poible with the CLS formulation to pecify region of the pf where delivered energy hould be maximized while at the ame time pecifying region where acoutic energy hould be minimized. In the LCLS deign cae, the linear contraint could be augmented to contrain the peak gain at a number of point location, which would reult in the T C vector of (6) becoming a matrix whoe row ize correponded to the number of hot pot. The QCLS algorithm may produce even better reult for thi cenario ince the quadratic contraint could be modified to include all region where the energy of the pf hould be contant. Many author have conidered the iue of improving the etimation of the blood flow velocity vector by modulating the acoutic beam in the azimuthal direction uing the receive apodization function and uing an autocorrelation etimator to determine the lateral velocity [42], [43]. Similar CLS formulation could be deigned in order to produce apodization profile that generate pf with modulation in the azimuthal direction a well a the elevation dimenion. 21

22 Thee apodization profile may be able to produce patial modulation frequencie higher than thoe previouly produced, reducing the variance of the lateral motion etimate [44]. DISCUSSION CLS apodization deign i a general formulation for deigning mathematically optimal ytem repone. We decribe formulation of thi approach for a variety of imaging application. The required propagation function can be determined through experiment, imulation, or theory. Implementing CLS apodization i conceptually and practically imple. Apodization weight could be pre-calculated and tored for the intended application, then retrieved from a look up table during imaging. Current ytem already employ dynamic apodization, o implementing the CLS profile on clinical canner hould be traightforward. However, in order for CLS apodization to be implemented on a clinical ytem, thorough characterization of the ytem i required, including the hift variance of the ytem repone. Depending on the ytem the degree of patial variance in the impule repone may be great or could be neglected. Either way, once the ytem ha been well characterized our algorithm ave a great deal of development time by obviating iterative deign. For cyt imaging, ucceive imaging with apodization profile correponding to all different deign cyt radii would be impractical. The quetion become, which apodization profile hould be ued? Simulation reult, preented in an accompanying paper [2], how that CLS apodization profile are relatively table acro a range of cyt ize. In fact, profile calculated for pecific cyt radii outperform conventional window at all cyt ize. Chooing 22

23 the appropriate apodization profile i traightforward when analyzed with the cytic reolution metric. Each apodization yield a contrat curve a a function of cyt radiu, and therefore chooing the optimal profile imply require electing the apodization that achieve a pecified level of contrat for the mallet cyt, or chooing the profile that yield the bet contrat for a given cyt ize. The author acknowledge that the reolution metric, while greatly improving theoretical deign conideration, till ha ome hortcoming. The mot worriome i that the metric decribe contrat at a given point in pace at a pecific intant in time. The metric quantifie contrat of the cyt center veru the background, not the overall cytic contrat. Incorporating detection algorithm, where contrat i defined relative to a peckle region, uch a thoe preented in [45] and [46] may be neceary. Thi notion raie further quetion regarding cyt detectability and oberver efficiency [47]. Which cyt i eaier to detect: (1) a cyt with well defined boundarie but low overall contrat, or (2) a cyt with blurred edge but greater maximum contrat? A imple one dimenional analyi yield ome inight (Fig. 2). If we view image formation a a imple convolution between the impule repone of the imaging ytem and the target function, then the CLS algorithm produce two very different reult. In one dimenion the cyt i modeled a a rect function ubtracted from a contant i.e. (1 rect). In the LCLS cae, the pf may reemble a triangle function whoe bae correpond to the width of the cyt. The reulting image of thi cyt will be a mooth Gauian like function. Note that in Fig. 2 that the reulting convolution are not to cale with the original input. In the QCLS cae, the cyt remain the ame but the pf reemble a rect function whoe width correpond to the ize of the cyt. The reulting image of the cyt will be a triangle function whoe negative peak i deeper than that of the mooth Gauian reulting from the LCLS 23

24 apodized pf. It i not obviou which cyt would be more readily detectable. One ha harper edge (LCLS) but the other ha a greater maximum contrat. Thi effect wa alo een in Johnon analyi comparing Hamming and flat apodization [32]. We hope to explore thi iue through a human oberver tudy where detection of cyt uing the CLS apodization profile will be invetigated. Fig. 2. One dimenional repreentation of imaging a cyt with the two CLS pf. In the LCLS cae the reulting image of the cyt ha harper defined edge. In the QCLS cae the reulting image ha blurred edge but more overall contrat than the LCLS imaged cyt. The deign method decribed here i different than that decribed in [24] for everal reaon. Firt, we do not require a goal pf in our minimization. Second, our method i coniderably eaier to implement for broadband imaging ytem, while till taking into account the 4 dimenional patio-temporal nature of the pf. Another difference i that the algorithm decribed by Ranganathan ha no contraint on the deign weight which could lead to intability of the reult. Our algorithm i imilar to the MSSE method decribed by Ranganathan in that it i general enough to apply to both one way and two way repone, continuou wave and broadband operation, and can be ued to deign aperture for a variety of 24

25 application. Another imilarity i that the entire pf i ued to obtain a leat quare olution to an overdetermined ytem of equation. Our method of apodization profile deign for coherent imaging ytem i imilar to many previouly decribed array pattern ynthei technique. The LCLS formulation i imilar to the array pattern ynthei technique by Teng [8], however we ue a broadband formulation, a different linear contraint on the weight, and our algorithm doe not require iteration to achieve an optimum. Thi QCLS formulation i imilar to the contrained eigenfilter deign [10], [26]; however, we ue a broadband formulation and a quadratic contraint on the energy in the pf. Keitmann-Curde et al. [27] ued a formulation imilar to our QCLS formulation in deigning optimal apodization in imulated ultraound field. However, their method minimized the energy in the idelobe of the ytem repone over time. The ue of pace-time pf in their model neglect the inherent hift variant propertie of the imaging ytem and integrating the preure field power over the time axi i not a realitic meaure of the ytem patial impule repone. Finally Schwann et al. elegantly ued two different reolution criteria to deign frequency dependent receive apodization profile [28]. Their reolution criteria: maximum to average ratio and the fill-in meaure relate directly to cytic contrat. However, their multiple objective formulation require computationally expenive iterative method to arrive at one Pareto optimum olution. Furthermore, given a Pareto optimal frontier, a curve of all the Pareto optimal olution a a function of the objective, for the maximum to average ratio veru fill-in criteria, it i unclear a to what point on that curve i truly optimal for clinical ultraound imaging. The contrat curve we produce uing the cytic reolution metric on the other hand, provide a traightforward approach for parameter optimization. 25

26 Our method offer an elegant path for optimizing the ultraound ytem pf and ha potential application to apodization deign for many varied application. Overall, the CLS apodization deign technique ha the potential to improve the contrat of anechoic legion and improve beamforming in general by forming pf with narrow mainlobe and low idelobe. The technique may alo aid in the deign of ytem repone ued for hyperthermia application and Doppler ignal proceing. Stability of the CLS algorithm in the preence of ound peed error a well a changing ytem parameter need to be invetigated. Simulation reult addreing thee iue for the CLS apodization deign technique and it application are decribed in an accompanying paper [2]. Reult how that CLS apodization profile improve cytic contrat compared to many conventional window a well a improve DOF. Concluion The CLS apodization deign technique preented in thi paper i a general beamforming method that can be ued to deign aperture for pecific application. It achieve mathematically optimal cytic contrat by deigning mathematically optimal aperture weight for a given ytem. The contrat i optimized becaue the weight minimize the energy of the pf outide the pecified cyt boundary while either maintaining peak gain inide the cyt (LCLS) or maintaining contant pf energy inide the cyt (QCLS). The CLS apodization deign technique alo ha the potential to improve point reolution by forming pf that have narrower mainlobe and lower idelobe than pf generated from conventional window. Therefore, we believe the CLS apodization deign algorithm have ignificant potential to improve ultraound beamforming and can be applied in any ultraound application where the ytem repone i well characterized. 26

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