ANALYSIS AND OPTIMIZATION OF SYNTHETIC APERTURE ULTRASOUND IMAGING USING THE EFFECTIVE APERTURE APPROACH. Milen Nikolov, Vera Behar

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1 International Journal "Inforation heories & Applications" Vol. 57 AALYSIS AD OPIMIZAIO OF SYHEIC APEUE ULASOUD IMAGIG USIG HE EFFECIVE APEUE APPOACH Milen ikolov, Vera Behar Abstract: An effective aperture approach is used as a tool for analysis and paraeter optiization of ostly known ultrasound iaging systes - phased array systes, copounding systes and synthetic aperture iaging systes. Both characteristics of an iaging syste, the effective aperture function and the corresponding two-way radiation pattern, provide inforation about two of the ost iportant paraeters of iages produced by an ultrasound syste - lateral resolution and contrast. herefore, in the design, optiization of the effective aperture function leads to optial choice of such paraeters of an iaging systes that influence on lateral resolution and contrast of iages produced by this iaging syste. It is shown that the effective aperture approach can be used for optiization of a sparse synthetic transit aperture (SA) iaging syste. A new two-stage algorith is proposed for optiization of both the positions of the transitted eleents and the weights of the receive eleents. he proposed syste eploys a 64-eleent array with only four active eleents used during transit. he nuerical results show that Haing apodization gives the best coproise between the contrast of iages and the lateral resolution. Keywords: Ultrasound iaging, Synthetic aperture, stochastic optiization. ACM Classification Keywords: J.3 Life and Medical Sciences: Medical inforation systes; I.5.4 Pattern recognition: Applications --- Signal processing; G..6 uerical Analysis: Optiization --- Siulated annealing. Introduction Medical ultrasound iaging is a technique that has becoe uch ore prevalent than other edical iaging techniques since this technique is ore accessible, less expensive, safe, sipler to use and produces iages in real-tie. However, iages produced by an ultrasound iaging syste, ust be of sufficient quality to provide accurate clinical interpretation. he ost coonly used iage quality easures are spatial resolution, iage contrast and frae rate. he first two iage quality easures (resolution and contrast) can be deterined in ters of bea characteristics of an iaging syste bea width and side lobe level. In the design of an iaging syste, the optial set of syste paraeters is usually found as a tradeoff between the lowest sidelobe peak and the narrowest bea width of an iaging syste. In a conventional ultrasound iaging syste, the transducer is a phased array with a great nuber of eleents (PA iaging systes). he quality of iages produced by a PA syste directly depends on the nuber of active channels used both in transission and receiving. hus, the conventional high-resolution PA iaging systes produce iages at relatively high cost []. Conventional phased array iaging systes eploy all eleents of the transducer during both transit and receive during each excitation cycle, while eploying delays in order to steer the bea and scan a D plane. In receive ode, dynaic (or coposite) focus is used, by adjusting the delays of transducer eleents as a function of the depth being iaged. In transit ode, usually the focus point is set in the iddle of the region being iaged. At the focus point, the lateral beawidth is the sallest (and the best lateral resolution is obtained there), while away fro the focus point, the lateral beawidth increases. he spatial resolution of the ultrasound iage can be iproved by using several transit beas during the interrogation of each sector, each of which is focused at a different depth. It is done in odern ultrasound iaging systes at the cost of decrease of the frae rate, proportionally to the nuber of transit foci []. An alternative way to obtain an appropriate spatial resolution, without the decrease of the frae rate, is to use the synthetic aperture technique. his ethod akes it possible to generate iages with dynaic focusing, during both transit and receive, while aintaining or even drastically decreasing the tie of iage acquisition. In a classical Synthetic Aperture Focusing echnique (SAF), only a single array eleent transits and receives at each tie. All the eleents are excited sequentially one after the other, and the echoes received are recorded

2 58 International Journal "Inforation heories & Applications" Vol. and stored in coputer eory. It reduces the syste coplexity and the frae rate, but requires data eory for all data recordings [3]. he ain disadvantage of SAF is the low signal-to-noise ratio (S) and as a result, the poor contrast resolution. In a Multi-eleent Synthetic Aperture Focusing (MSAF) ethod, at each tie a group of eleents transits and receives signals siultaneously [4]. he transitted bea is defocused to eulate a spherical wave. he S is increased copared to SAF, in which only a single eleent is used in transit and receive. In a Synthetic ransit Aperture (SA) ethod, at each tie one array eleent transits a pulse, and all eleents receive the echo signals [5]. Copared to conventional phased array iaging, the advantage of this approach is that a full dynaic focusing can be applied to the transission and the receiving, producing the highest quality of iages at the increased frae rate. he shortcoing is that a huge data eory is required for data recordings. For an -eleent array, echo recordings are required to for a conventional phased array iage, and, however, echo recordings are required to synthesize a SA iage. his disadvantage can be overcae to soe extent, if only a few eleents, M, act as transitters. In that case M echo recordings are required to synthesize a SA iage, where M< [6]. his is equivalent to using of a sparse array in transit. he sparse SA iaging acquires iages at higher frae rates, which akes this ethod very attractive for real-tie 3D-ultrasound iaging. he relation between the eployed effective aperture function and the resultant radiation pattern of the iaging syste can be used as a strategy for analysis and for optiisation of an iaging syste [7]. Since the two-way radiation pattern of a syste is the Fourier transfor of the effective aperture function, the transitted and receiving radiation patterns can be optiised by selecting the appropriate transit and receive aperture functions, to produce the desired effective aperture of the iaging syste. hus, when the desired effective aperture of a syste is defined, it also provides the two-way radiation pattern that should be used, with the appropriate width of the ain-lobe and its sidelobes. In synthetic aperture iaging, the transit aperture function depends not only on the nuber of transit eleents, but also on their geoetrical locations within the array (sparse synthetic aperture iaging). he received aperture function depends on the length of a physical array and the apodization weights applied to the receiver eleents. hus, the shape of the effective aperture function and, therefore, the shape of the two, one-way radiation patterns of a syste, can be optiised depending on the positions of the eleent in transit and the weights of the eleent in receive. In this paper, it is shown how the effective aperture approach can be used for analysis and paraeter optiisation of an ultrasound SA iaging syste. Using this approach, the optial set of syste paraeters (nuber of array eleents, their configuration within an array) can be deterined in result of a coproise between the lowest sidelobe peak and the narrowest bea width of the two-way radiation pattern of an iaging syste. he coparison analysis of 3 types of iaging systes is done calculating their effective aperture function and the corresponding two-way radiation pattern using the coputational environent of Matlab.. he Effective Aperture Concept he effective aperture of an array represents an equivalent aperture that would produce identical two-way radiation pattern if the transit aperture was a point source. An expression for the effective aperture of an array can be derived fro a calculation of the two - way radiation pattern. Consider an uniforly spaced linear array of eleents with weighting w() (,...,. ). he one- way far field bea pattern is W k d sin ( θ ) ( ) w( ) e θ () where d and k o are the inter-eleent spacing and the wave nuber, respectively. his equation can also be described as a discrete Fourier transfor (DF) of the aperture function: W ( k) DF[ w( ) ] ( ) w e π j k, k,,, - () in which the frequency index k aps into the bea angle θ by sinθ kλ/(d) where λ is the wavelength. Since the round-trip bea pattern is the product of the transit and receive beas

3 International Journal "Inforation heories & Applications" Vol. 59 using the DF property, we get ( θ ) W ( θ ) W ( θ ) W (3) W ( k ) DF[ w w ] (4) where denotes convolution and w and w are the apodization functions applied to the array eleents in transit and receive, respectively. Using (4), the effective aperture function of an iaging syste is defined as e w w and W ( θ ) FF ( e ) (5) hus, the round trip bea pattern is deterined by the transitted and the receiving aperture weightings. Every physical bea can be realized by foring the appropriate effective aperture. 3. Synthetic Aperture Iaging First, the concept of synthetic aperture was originally used in radar for highly resolution iaging terrain, but it can be successfully used in ultrasound iaging systes as well. In this case, the benefit of the synthetic aperture is the reduction of syste coplexity and cost. Several ethods were proposed to for a synthetic aperture for ultrasonic iaging. In SAF iaging, at each tie only a single array eleent transits a pulse and receives the echo signal. (Fig. ). he syste coplexity is reduced, because only a single set of circuit for transit and receive is needed. In this case the effective aperture can be calculated by e w ( ) w ( ), where w ( ) w ( ) [,,, i, K,] and i (6) K In MSAF iaging, a group of eleents transit and receive signals siultaneously, and transit bea is defocused to eulate a single eleent response (Fig. ). he acoustic power and the signal-to-noise ratio are increased copared to SAF where a single eleent is used. his ethod requires also eory for data recordings. In MSAF, a Kt-eleent transit subaperture sends an ultrasound pulse and echo signals are recorded at a Kr-eleent receive subaperture. At the next step, one eleent is dropped and a new eleent is included to the transitted and receiving subaperture, repeating the transission and receiving process. Usually K t K r k. he effective aperture is: k + ( ) w ( ) e w, where (7) ( ) w ( ) [,, K, i, i, K, i,,,] w and i i... i k K + k ASMI MODE FIIG ECEIVE MODE SEPS i ASMI MODE FIIG ECEIVE MODE SEPS i -k+ One active eleent One active eleent k active eleents k active eleents Fig.: SAF iaging ethod Fig.: MSAF iaging ethod In SA iaging, at each tie one array eleent transits a pulse and all eleents receive the echo signals (Fig. 3). he advantage of this approach is that a full dynaic focusing can be applied to the transission and

4 6 International Journal "Inforation heories & Applications" Vol. the receiving, giving the highest quality of iage. he disadvantage is that a huge data eory is required and otion artifacts ay occur. he effective aperture is calculated as: e w w ( ) where w [,, K,], ( ) [,,, i, K,] w and i (8) K Synthetic eceive Aperture (SA) ethod of iaging was proposed to iprove lateral resolution (Fig. 4). It is known that the lateral resolution can be iproved by increasing array length. In practice, it is not very expensive to build a large transit aperture, but is very coplex to for a large receive aperture. his ethod uses a large transit aperture and enables an iaging syste to address a large nuber of transducer receive eleents without the sae nuber of parallel receive channels. In the receive ode the aperture is split into two or ore subapertures. In order to for each line of iage data in the SA syste, the transitters ust be fired once for each receive subaperture. For a single transit pulse (fro all transit eleents), the F su for one receive subaperture is fored and stored in eory. hen a second identical pulse is transitted in the sae direction and the F su for another subaperture is fored and stored. After the F signals have been acquired fro all receive subapertures, the total F su is fored by coherently adding together the sus fro various subapertures. For an -eleent linear array, receive aperture is split into S /K subapertures, and each subaperture contains K eleents. he effective aperture is: S ( ) e w w, where (9) w [,, K,], ( ) [,, K, i, i, K, i,,] n in... in+ K i + w, n * K + n n+ n+ K, K ASMI MODE FIIG ECEIVE MODE SEPS i ASMI MODE FIIG ECEIVE MODE SEPS... i S One active eleent All active eleents All active eleents K active eleents Fig.3: SA iaging ethod Fig.4: SA iaging ethod A sparse SA iaging ethod is proposed to increase syste frae rate (Fig. 5). Only a sall nuber of eleents are used to transit a pulse but all array eleents receive the echo signals. For an -eleent aperture, Mx data recordings are needed for iage reconstruction., where M<<. All data recordings ust then be cobined with dynaic focusing. he effective aperture is: e M where w ) [,,... i,..], w [,,...,] and i ( K w ( ) w () he two-way radiation pattern of a synthetic aperture iaging syste is calculated using the Fourier transfor of the corresponding effective aperture function defined by the expressions (6,7,8,9 and ). K

5 International Journal "Inforation heories & Applications" Vol Optiization of a Sparse Array For a sparse SA iaging syste with an array with -eleents, the two-way radiation pattern is evaluated as the Fourier ransfor of the effective aperture function e, defined as: e M a B, and a,,..., i,...], where i () [ where a is the transit aperture during the th firing, B is the apodization function applied to the receiver eleents, and is the convolution operator. he speed of the iage acquisition is deterined by the nuber of transit eleents (M), M<<. Since the geoetrical locations of the transit eleents in a sparse array syste ipact the two-way radiation pattern of that syste, the iage quality paraeters, the lateral resolution and contrast, all depend on the locations of the transit eleents within the sparse array (i,i,,i M ). Since the weighting applied to each receiver eleent also ipacts the radiation pattern of the syste, the iage quality also depends on the type of the apodization function (B). herefore, the optiization of a sparse SA iaging syste can be forulated as an optiization proble of both the location of the eleents of the sparse array in transit and the weights assigned to the eleents of the full array during receive [8]. Different algoriths have been proposed for optiization of the locations of the transitted eleents in a sparse array genetic, linear prograing and siulated annealing algoriths [8]. For ost cases the optiization criterion is inial sidelobe peak of the radiation pattern. In this paper, another optiization criterion is proposed. It is the inial width of the ainlobe (W) cobined with a condition on the axiu sidelobe level (SL <Q). It is suggested here to divide the optiization process into two stages. In the first stage, the optial positions of transit eleents ((i,i,,i M ) are found, for a set of known apodization functions {B k }, k,, K. Such a set of apodization functions ay include several well-known window-functions (Haing, Hann, Kaiser, Chebyshev and etc). At this stage, the optiization criterion can be written as follows: Given M, and {B} K, choose (i,i,,i M ) K to iniize W subject to SL<Q () where Q is the threshold of acceptable level of the sidelobe peak. In the second stage, the final layout of transit eleents is chosen, which is a layout that corresponds to the ost appropriate apodization function B (b,b,..,b ). his choice is a coproise between inial width of the ainlobe and the acceptable level of the peak of the sidelobes. Matheatically, it can be written as follows: Given M,, {B} K and {i,i,,i M } K, choose (b,b,..,b ) to iniize W subject to SL<Q (3) ASMI MODE FIIG SEPS... i One active eleents M ECEIVE MODE Fig.5: Sparse SA iaging ethod All active eleents begin Initialize (I, ) for k to nuber_iterations k k α for j to nuber_perturbations Ipperturbate(I j- ) ΔWW(Ip)- W(I j-) ΔSLSL(Ip)- Q if {ΔW< or exp(-δw/ k )<rand(,)} & {ΔSL< or exp(-δsl/ k )<rand(,)} I j Ip else I j I j- endif endfor endfor end Fig.6: he siulated annealing algorith

6 6 International Journal "Inforation heories & Applications" Vol. where {i,i,,i M } K are the selected positions of transit eleents, as found at the first stage of the optiization. One way of selecting the positions {i,i,,i M } K is by using a odification of the siulated annealing algorith based on a Monte Carlo siulation. his approach was suggested initially for cobinatorial optiization by Kirkpatrick et al. [9]. he siulated annealing algorith realizes an iterative procedure that is deterined by siulation of the arrays with variable transit eleent positions. In order to speed up the siulation process it is assued that two of the M transit eleents are always the two outer eleents of the physical array; their positions are not changed and are assigned nubers and. he positions of the other transit eleents are shifted randoly, where a shift in position to the left or to the right has equal probability (of.5). Once the process is initiated, with an initial layout of transit eleents I (i, i,..i M), a neighbour layout I (i, i,..i M) is generated, and the algorith accepts or rejects this layout according to a certain criterion. he acceptance is decided stochastically and ay be described in ters of probability as: if ΔW < & ΔSL < exp( ΔW / k ) if ΔW > & ΔSL < P (4) exp( ΔSL / k ) if ΔW < & ΔSL > exp( ΔW / k ) exp( ΔSL / k ) if ΔW > & ΔSL > where P is the probability of acceptance, ΔW is the difference of width of the ainlobe, ΔSL is the difference of the height of the peak of the sidelobe between the current configuration of transit eleents and the best one obtained at preceding steps. k is the current value of teperature, where the current teperature is evaluated as k.95 k-, and the algorith proceeds until the nuber of iterations reaches the final value. A pseudo-code of the proposed siulated annealing algorith is given in Fig Coputations and Coparison Analysis he effective aperture function and the corresponding two-way radiation pattern of several ultrasound iaging systes were calculated using Eqns (-4), in order to copare the quality of iages produced by the questioned systes. Synthetic aperture. hree ore perspective types of synthetic aperture iaging systes are investigated. he investigated MSAF iaging syste eploys a linear array with 64 eleents and active sub-apertures with 4, 8 (Fig. 7), 6 (Fig. 8) or 3 eleents, respectively. In the study no apodization is used. he best results for the lateral resolution and SL are obtained when the active transit sub-aperture consists of only 4 eleents (able ). he ain disadvantage of the sall nuber of active transit eleents is that the transitted power is less, hence the S is low. Fig. 7: MSAF ethod with 8 active eleents Fig. 8: MSAF ethod with 6 active eleents

7 International Journal "Inforation heories & Applications" Vol. 63 Fig.9 Conventional SA ethod (64-eleents, Haing apodization) Fig. : Sparse SA ethod (4 transit eleents, 64 receive eleents, Haing apodization) uber of transit eleents Δθ SL, db able : Mainlobe and sidelobe peak level of a MSAF iaging syste In a conventional SA iaging ethod, the transducer is of 64 eleents. In the receive ode all eleents are active. In the transit ode, the linear array is split into 64 sub-apertures, each of the has only one active eleent. he effective aperture function and the corresponding two-way radiation pattern of such a SA iaging syste is shown in Fig.9. he 6dB beawidth obtained for the SA iaging syste is.8º. However, the sidelobe peak level of the SA syste is only 3 db. Sparse array optiization. Coputer siulations were perfored in order to optiize the design and perforance of a sparse array probe, to be used for synthetic transit aperture iaging. he exaple given here is of a 64 eleents sparse array, where 64 active eleents are used in receive and only 4 eleents are used in transit. he properties of the syste are optiized using the two-stage algorith described in section 4. First, the optial positions of transit eleents are found for three apodization functions Boxcar (i.e. no apodization of the receiver eleents), Haing and the Blackan-Harris. Optiized positions of transit eleents eceiver Apodization Mainlobe width (Δθ ) -6 db - db -4 db SL (db),, 63, , 6, 39, 64 Haing ,, 44, 64 Blackan- Harris able : uerical results obtained after eploying the two stages of the optiization. For each apodization function, the positions of transit eleents are shifted until optial perforance is obtained, as described earlier, using the siulated annealing algorith presented in Fig.6. In order to obtain a radiation pattern with a sharper ainlobe, the optiization criterion was forulated as the inial width of

8 64 International Journal "Inforation heories & Applications" Vol. the ainlobe at db ( instead of at 6 db) below the axiu where the condition that the axial level of the sidelobe peak is below 5 db. he positions of transit eleents that were found to optiize the perforance of the syste, studied for a physical array with λ / eleent spacing, together with the achieved widths of the ainlobe (at -6 db, - db and 4 db) and the levels of the peaks of the sidelobe, are all presented in able. Both optiized functions, the effective aperture function and the corresponding two-way radiation pattern, are plotted for Haing apodization function (Fig. ). It ay be seen that the apodization reduces the levels of the peaks of the sidelobes fro 33 db to db, but at the cost of widening the ainlobe of the radiation pattern. Since the dynaic range of a coputer onitor is liited to about 5 db, the sparse array is chosen with the Haing apodization and the locations of the transit eleents are set to be at positions, 6, 39 and 64. Coparison analysis of nuerical results (able, Fig.9) shows that a sparse SA iaging syste iproves significantly lateral resolution of iages because Δθ.8º - for a conventional SA iaging syste and Δθ.34º - for a sparse SA iaging syste. 6. Conclusions It is shown that the effective aperture approach can be successfully used as a tool for analysis and paraeter optiization of the synthetic aperture iaging systes. he effective aperture function and the corresponding two-way radiation function provide inforation about two of the ost iportant paraeters of iages produced by an ultrasound syste - lateral resolution and contrast. herefore, in the design, optiization of the effective aperture function leads to optial choice of such paraeters of a SA iaging syste that influence on lateral resolution and contrast of iages produced by this iaging syste. he nuerical results show that each syste has its own advantages and disadvantages. he choice of iaging syste should depend on the task, which it will be used for. It is shown that Haing apodization gives the best coproise between the contrast of iages and the lateral resolution. A MSAF syste has better lateral resolution and SL level with less active eleents, but in that case the S is lowered. he sparse synthetic transit aperture iaging systes can be proposed as an alternative and superior approach to the conventional SA systes. Yet, the sparse SA iaging systes suffer fro soe deficiencies. With proper design, these deficiencies can be overcoe and the sparse SA iaging syste can perfor extreely well for specific applications. o do so, an effective aperture approach is used for optiization of the sparse SA iaging syste. A two-stage algorith is proposed for optiizing both the locations of transit eleents within the ultrasound probe and the weights of the receive eleent. he first stage of the optiization procedure eploys a siulated annealing algorith that optiizes the locations of the transit eleents for a set of apodization functions. At the second stage, an appropriate apodization function is selected. Acknowledgents his work was supported by the Centre of Excellence BIS++ and the Bulgarian ational Science Fund grants I-/, I-5/ and MI-56/5. Bibliography B. Angelsen, Ultrasound iaging: Waves, signals, and signal processing, Eantec, orway,. S. Hol and H. Yao, Method and apparatus for synthetic transit aperture iaging, US patent o , Sep. 4, 999. Ylitalo, On the signal-to-noise ratio of a synthetic aperture ultrasound iaging ethod, Europ. J. Ultras. 3, (996), M.Karaan, H. Bilge, and M. O Donnell, Adaptive ulti-eleent synthetic aperture iaging with otion and phase aberration correction, IEEE rans. Ultrason. Ferroelec. Freq. Contr., vol. 45, 4, (998), G. rahey, and L. ock, Multi-eleent synthetic transit aperture iaging using teporal coding, IEEE rans. Med. Iag., vol., 4, (3), V. Behar, and D. Ada, Optiization of sparse synthetic transit aperture iaging with coded excitation and frequency division, Ultrasonics, (5), (subitted to be printed) G. Lockwood and F. S. Foster, Design of Sparse Array Iaging Systes

9 International Journal "Inforation heories & Applications" Vol. 65 S. Hol, A. Austeng, K. Iranpour, J. Hopperstad, Sparse sapling in array processing, Chapter 9 in "Sapling theory and practice, (F. Marvasti Ed.), Plenu,.Y., () S. Kirkpatrik, C. Gelatt, M. Vecchi, Optiization by siulated annealing, Science, vol., 4598, (988), Authors' Inforation Milen ikolov Institute for Parallel Processing, Bulgarian Acadey of Science, Acad. G. Bonchev Str., 5-A, Sofia 3, Bulgaria, e-ail: ilenik@bas.bg Vera Behar Institute for Parallel Processing, Bulgarian Acadey of Science, Acad. G. Bonchev Str., 5-A, Sofia 3, Bulgaria, e-ail: behar@bas.bg A MAHEMAICAL APPAAUS FO OOLOGY SIMULAIO. SPECIALIZED EXESIOS OF HE EXEDABLE LAGUAGE OF APPLIED LOGIC Alexander Kleshchev, Irene Artejeva Abstract: A atheatical apparatus for doain ontology siulation is described in the series of articles. his article is the second one of the series. It describes a few specialized extensions of the extendable languages of applied logic that was described in the first article of the series. A few exaples of soe ideas related to doain ontologies and foralization of these ideas using the language are presented. Keywords: Extendable language of applied logic, ontology language specification, specialized extensions of the extendable language of applied logic. ACM Classification Keywords: I..4 Knowledge epresentation Foraliss and Methods, F4.. Matheatical Logic Introduction he definition of the extendable language of applied logic was given in [Kleshchev et al, 5]. his definition consists of the kernel of the language and of its standard extension only. When the seantic basis is extended for particular applications the following two classes of eleents are possible. he eleents of the first class can be ipossible or undesirable to be defined by eans of the kernel of the language and by extensions built. On the contrary, the eleents of the second class can be naturally defined by eans of the kernel and extensions built. he eleents of the first class are described in specialized extensions in the sae for that is used in the description of the kernel of the language and of its standard extension. A specialized extension of the language defines eleents of the seantic basis that are necessary for a coparatively narrow class of applications. Because the sae specialized extensions can be used in different applications such extensions have naes. Every particular language of applied logic contains the kernel and usually the standard extension and possibly soe specialized extensions. By this eans, every particular language of applied logic is characterized by a set of extension naes rather than a signature. A signature is introduced by a particular logical theory represented by his paper was ade according to the progra of fundaental scientific research of the Presidiu of the ussian Acadey of Sciences «Matheatical siulation and intellectual systes», the project "heoretical foundation of the intellectual systes based on ontologies for intellectual support of scientific researches".