capon beamforming in medical ultrasound imaging with focused beams

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 55, no. 3, march 2008 619

Capon Beamforming in Medical UltrasoundImaging with Focused Beams

Francois Vignon and Michael R. Burcher

Abstract—Medical ultrasound imaging is convention-ally done by insonifying the imaged medium with focusedbeams. The backscattered echoes are beamformed usingdelay-and-sum operations that cannot completely eliminatethe contribution of signals backscattered by structures offthe imaging beam to the beamsum. It leads to images withlimited resolution and contrast.

This paper presents an adaptation of the Capon beam-former algorithm to ultrasound medical imaging with fo-cused beams. The strategy is to apply data-dependentweight functions to the imaging aperture. These weights actas lateral spatial filters that filter out off-axis signals. Theweights are computed for each point in the imaged medium,from the statistical analysis of the signals backscattered bythat point to the different elements of the imaging probewhen insonifying it with different focused beams.

Phantom and in vivo images are presented to illustratethe benefits of the Capon algorithm over the conventionaldelay-and-sum approach. On heart sector images, the clut-ter in the heart chambers is decreased. The endocardiumborder is better defined. On abdominal linear array im-ages, significant contrast and resolution enhancement areobserved.

I. Introduction

The standard algorithm currently used by most com-mercial ultrasound scanners is the delay-and-sum

(DAS) beamformer. To estimate the scattering strength ofone point of the imaged medium, an array of piezoelectricelements first sends out a focused wave into the medium inthe direction of the point of interest. The waves backscat-tered by the medium are recorded by the array. Out ofthis spatio-temporal data, only the data samples that cor-respond to the time for a wave to travel from the arrayto the point of interest and back from that point to thedifferent elements of the array are kept (delay step). Thesesamples are then summed over the elements (sum step) togive the value of the image pixel that corresponds to thatpoint of interest.

The resolution and contrast achievable with thismethod are fundamentally limited. Indeed, the times atwhich signals backscattered by the point of interest reachthe different elements of the imaging array (the delaycurve) intersect with delay curves corresponding to neigh-boring points. The data samples corresponding to the sig-nal backscattered by the point of interest (on-axis signals)

Manuscript received July 20, 2007; accepted December 17, 2007.The authors are with Philips Research North America, Ul-

trasound Imaging and Therapy, Briarcliff Manor, NY (e-mail:[email protected]).

Digital Object Identifier 10.1109/TUFFC.2008.686

are thus loaded with signals backscattered by neighboringpoints (off-axis signals). As a consequence, a given pixel’svalue does not just correspond to the backscattering prop-erties of the physical point it represents but is corruptedby the backscattering properties of neighboring points. Apoint scatterer thus always appears, on the image, to havea finite size and shape, called the point spread function(PSF). The PSF depends on the size of the array, the work-ing frequency, and the position of the imaged point withrespect to the imaging probe, according to the diffractiontheory. The PSF is usually described by the width of itsmain lobe and the strength of its sidelobes. These param-eters measure how much a pixel’s value can be corruptedby neighboring points.

In medical images, it is thus sometimes difficult to dis-tinguish pointlike objects in a highly scattering environ-ment like biological tissue. Furthermore, void areas in theimaged medium can erroneously appear with significantlyhigh pixel values on the image, corresponding to signalbackscattered by neighboring tissue insonified by the side-lobes of the PSF (the clutter effect).

The aim of the Capon beamforming algorithm [1] isto reduce the contribution of off-axis signals to the pixelvalue. The approach is to weight the received per-elementdelayed data samples prior to summing them. This delay-weight-and-sum approach can be seen as applying a spatialfilter to the signals received across the probe aperture tofilter out the off-axis contributions [2]. New weights arecomputed for each point of the imaged medium. Indeed,the off-axis contributions that have to be filtered out de-pend on the PSF of the point of interest and the scatterersinsonified by this PSF. The weights are computed fromthe statistical analysis of the signals corresponding to thepoint of interest (and, inevitably, to its neighborhood) overseveral “observations” of that point.

Several groups have presented implementations of theCapon algorithm to ultrasound imaging with focusedbeams by taking the backscattered data at different timesas the different observations of one point [3]–[5]. However,to avoid mixing information from points at different depths(a later observation corresponds to data backscattered by adeeper point) only a few data samples can be taken into ac-count. Furthermore, the original Capon algorithm requiresthe different observations of the medium to yield uncorre-lated data. Using different data samples separated in timeas different observations, the observations are naturallycorrelated because the backscattered data is composed ofreplicas of the same imaging waveform. To overcome thisproblem, a spatial smoothing preprocessing scheme [6] has

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620 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 55, no. 3, march 2008

to be used, as in [4] and [5]. Alternatively, knowledge of theimaging system’s frequency response can be introduced (asin [3]) at the cost of an increased computational load (thisspecific extension of the Capon beamformer to broadbandsignals is known as the Frost beamformer [7]).

Prada et al. [8] and Wang et al. [9], in their implemen-tations of the Capon beamformer to ultrasound imaging,perform the different observations of the medium by in-sonifying it with different beams. This approach aims tolimit the correlation between the different observations. Inthose works, the beams are generated by single-elementtransmits in a synthetic aperture scheme, resulting in di-verging wavefronts. There are as many observations of agiven point as array elements. Although the algorithm ofPrada et al. works in the frequency domain and does notenable range resolution, the algorithm of Wang et al. worksin the time domain and can be applied to medical ultra-sound imaging.

The approach presented in this paper is similar to Wanget al. in that the different observations of the medium areprovided by different insonifications. The fundamental dif-ference lies in the fact that here, focused beams are usedto insonify the medium. Several practical advantages areexpected by using focused beams instead of cylindrical di-verging beams, such as a higher signal-to-noise ratio, thepossibility to apply the algorithm to harmonic data (har-monics are more efficiently generated with focused beams),and a reduced sensitivity to motion. For the aforemen-tioned reasons, most commercial ultrasound scanners usefocused beams for image formation, and thus an imple-mentation of the Capon beamformer with focused beamswould be more readily implementable than with a syn-thetic aperture acquisition.

The outline of the paper is the following: first, this newimplementation of the Capon beamformer with focusedbeams is described. Then, phantom and in vivo images ob-tained with the DAS and the Capon beamformers and dif-ferent array geometries (phased and linear) are displayed.The images are discussed in terms of the clinical signifi-cance of the observed changes.

II. The Capon Beamformer with Focused Beams

A. Several Observations of the Medium Through the Useof Focused Beams

In this paper, the image formation is described in apixel-wise basis. Each point P of the imaged medium hasa corresponding pixel value A(P ) in the image. The com-putation of the pixel value from the data is explained forone point P ; the operation has to be repeated for all pointsto have a complete image of the medium.

The first step in the adaptive beamforming algorithmis to perform several observations of each point P in themedium. Here, the different observations are realized byinsonifying the medium with different focused beams andeach time isolating from the backscattered data only what

comes from point P (this process is called “focusing inreceive” on point P ). If K different focused beams areused to insonify the medium, K realizations of the signalsbackscattered by point P (K “observations” of point P )are available.

1. One Observation of Point P : Most commercial ultra-sound scanners sequentially transmit focused beams intothe imaged medium. These focused beams differ by theirposition in space and hence the portion of the mediumthey insonify.

To perform one observation of point P , one of these fo-cused beams (beam number k) insonifies the medium, andthe backscattered signals are recorded by the N elementsof the imaging array. If T temporal samples are collectedon each element, the backscattered data can be formalizedby a T × N matrix Pk whose ith column ρk

i contains theT temporal data samples of the signal backscattered toelement i when insonifying the medium with transmit k:

Pk = [ρk1 . . . ρk

i . . . ρkN ]. (1)

The time samples that correspond to what has beenbackscattered by point P only, have to be extracted fromthis spatio-temporal data (focusing in receive on point P ).Assuming that the propagation medium is homogeneouswith speed of sound c, this can be done by geometricalconsiderations. Two different situations need to be consid-ered: point P is deeper than the focal depth, or point Pis shallower than the focal depth, as explained below andillustrated in Fig. 1.

The time when the central element of the array emitsis taken as the time origin. Let us call d1 the distancebetween the central element of the array and the focalpoint F of the transmit beam; d2 the distance between thefocal point and point P ; and d3 the distance between pointP and element i of the array. P is not necessarily on theaxis of the beam. The time τ(P, i, k) at which the signalbackscattered by point P arrives at element i followingtransmit number k, if P is deeper than F is

τ(P, i, k) =1c

(d1 + d2 + d3) . (2a)

And, if P is shallower than F :

τ(P, i, k) =1c

(d1 − d2 + d3) . (2b)

Indeed, if P is deeper than F , the time for the wavefrontto arrive at P is the time for it to converge at F andthen to propagate further to point P . If P is shallowerthan F , the wavefront arrives at P , then has to travel thedistance between P and F to converge. Thus the time forthe wavefront to arrive at P is the time for it to converge atF minus the propagation time from P to F . In each case,the backscattered signal still has to travel the distancefrom P to element i before being recorded by element i(see Fig. 1).

The delays τ(P, i, k) are used to select from the per-element temporal data in Pk only the temporal samples

vignon and burcher: capon beamforming in medical ultrasound with focused beams 621

Fig. 1. The delays to select what comes back from point P wheninsonifying the medium with transmit number k. On each figure arerepresented the kth focused beam, its focus F , the point of interest P ,and the emitted wavefront at the time it arrives at point P . Left: P isdeeper than F . Right: point P is shallower than F . The gray arrowson both figures indicate the propagating distances that have to becounted positively in the travel time computation, and the dashedarrow indicates the distance that has to be counted negatively.

that correspond to the time for the wave to travel fromthe array to point P and back to the different elementsof the array. One time sample is selected per element toyield the per-element delayed sample vector, or “receivevector” rk(P ) =

[rk1 (P ) · · rk

i (P ) · rkN (P )

]T correspond-ing to point P and transmit k. Its ith element is definedas follows:

rki (P ) = ρk

i (τ(P, i, k)). (3)

The receive vector rk(P ) corresponds to the kth “obser-vation” of point P .

2. Several Observations of Point P : To gain more in-formation about P and its neighborhood, one has to makeseveral observations of it. The observations are made byinsonifying the medium with K different focused beamsand for each, focusing in receive on point P . In the end,K “receive vectors” are available, corresponding to pointP : one receive vector per transmitted focused beam.

To have as much information as possible about P witha given number of K transmit events, the focused beamshave to just satisfy the Nyquist spatial sampling require-ment. The K beams are chosen symmetrically about thelocation of P , and spaced so that their lateral main lobesat focus are just independent in the Rayleigh sense: Twoconsecutive transmit beams should be spaced by half atransmit beam main lobe’s width.

B. The Delay-and-Sum (DAS) and the Capon BeamformerPixel Value Through One Particular Observation ofPoint P

The DAS pixel value corresponding to point P is ob-tained through one particular observation of point P . Forthis observation, the transmit beam is chosen so that ithas P on its axis. This transmit is indexed k(P ). The cor-responding receive vector rk(P )(P ) is extracted from theper-element temporal data matrix Pk(P ) through the useof the appropriate delay curve τ(P, i, k(P )), as describedin (1)–(3).

These per-channel delayed samples are then simplysummed over the array elements to give the DAS pixelvalue ADAS(P ). This sum operation can be formalized bya scalar product between the receive vector rk(P )(P ) and1N , a column vector of size N containing only ones:

ADAS(P ) = 1TNrk(p)(P ). (4)

The Capon beamformer pixel value ACB(P ) differsfrom the DAS pixel value in that the per-channel de-layed samples are weighted with pixel-dependent weightsprior to summation. Introducing the weight vector w(P ) =[w1(P ) · · wi(P ) · wN (P )]T , this weight-and-sum opera-tion can be formalized as a scalar product:

ACB(P ) = w(P )T rk(p)(P ). (5)

The next section explains how to compute the weightvector w(P ) from the available observations.

C. The Capon Weight Vector

1. Problem Definition and Computation of the Weights:The aim is to find the weight vector w(P ) that minimizes,on average over the observations, the contribution of off-axis signals to the beamformed energy. At the same time,w(P ) must correctly restitute the on-axis energy. This isa minimization under constraint problem that can be for-malized as follows:

w(P ) = arg minw

wT R(P )w (6)

with

w(P )T 1N = 1. (7)

In (6), the sample covariance matrix R(P ) has beenintroduced:

R(P ) =1K

K∑

k=1

rk(P )rk(P )T . (8)

R(P ) is thus built out of the aligned data coming frompoint P when insonifying it with different transmit beams.It is an N×N symmetric matrix. Its element R(P )i,i′ is thecorrelation coefficient over the transmit events of what isreceived by elements i and i′ coming from point P . Further


in the manuscript, it will be referred to as the “covariancematrix” for simplicity, but one should keep in mind thatit is only an estimate of the true covariance matrix.

In reality, (6) states that w(P ) should minimize, onaverage over the K observations, the whole beamformedenergy

(w(P )T rk(P )

)2 and not only the contribution ofoff-axis signals to it. This approximation to the ideal min-imization of off-axis energies has to be made because thereceive vectors corresponding only to off-axis signals arenot available, and the only measurable signals rk(P ) areinevitably a mix of on-axis and off-axis signals [10]. How-ever, the constraint of (7) ensures that the contribution ofon-axis signals to the pixel value is preserved. The contri-bution of off-axis signals to the pixel value is thus mini-mized.

The minimization under constraint problem describedin (6) and (7) has an analytical solution that can be derivedusing the Lagrange multiplier methodology [11]:

w(P ) =R(P )−11N

1TNR(P )−11N

. (9)

Once w(P ) has been computed, the Capon pixel valuefor point P is computed by applying the weights to thereceive vector rk(P )(P ) as described by (5).

2. Pseudo-Inversion of the Covariance Matrix: A crit-ical step in the implementation of the Capon beamformeris the inversion of the covariance matrix R(P ) in (9). In-deed, there are several reasons why R(P ) is ill-conditioned.First, if the number of observations that are made of thepoint of interest is less than the number of elements of thearray (K < N), then the rank of R(P ) is at most K andR(P ) is not invertible as is. Indeed, R(P ) can be writtenas a matrix product between a N × K data matrix Y(P )whose K columns are the K observations of point P , andits transpose (R(P ) = Y(P )Y(P )T ). The rank of a rect-angular matrix is at most the min of its number of columnsand its number of rows—rank(Y(P )) ≤ min(K,N)—andthe rank of a matrix product is at most the rank of thelowest-rank matrix in the product [12]. The rank of R(P )is thus at most K if K ≤ N . Furthermore, if the K differ-ent observations—the different receive vectors rk(P )—arenot completely independent, the rank of R(P ) is strictlyless than K. This happens when the waves backscatteredby the point of interest do not depend much on the insoni-fying beam, for example, when the point of interest is astrong point scatterer, or partially developed speckle.

To ensure a robust pseudo-inversion of R(P ), its diago-nal is first “loaded” with a point-dependent load λ(P ):A diagonal matrix λ(P )I, where I is the N × N iden-tity matrix, is added to R(P ). Here, λ(P ) is chosen asa function of the vectors and eigenvalues of R(P ), as de-scribed in Li et al. [13]. The user can tune the way λ(P )is computed toward more robustness for the algorithmor more efficient off-axis energy rejection. Then, only theK strongest eigenvalues of the diagonal-loaded covariancematrix R(P ) + λ(P )I are inverted, to take into account

the intrinsic rank-deficiency of R(P ) that comes from thelimited number of observations. The remaining eigenval-ues are not inverted and replaced by zeros in the inversematrix.

III. Materials and Methods

This version of the Capon beamformer has been im-plemented using a phased array (P4-2) and a linear ar-ray (L12-5 38 mm) and a modified version of the HDI5000 scanner (Philips Healthcare, Andover, MA) that al-lows us to access per-element temporal data. Phantom andin vivo per element data were processed off-line with theCapon beamformer and the traditional DAS beamformerin Matlab (The Mathworks, Natick, MA) to generate thecorresponding images. The DAS beamformed images areused as the reference in terms of image quality. The phan-tom used in these experiments is the multi-tissue phantom#040 (CIRS Inc., Norfolk, VA). It is made of bright inclu-sions, voids, and point scatterers, included in a specklybackground.

The P4-2 phased array is a broadband transducer witha 4 to 2 MHz extended operating frequency range. Its foot-print is 21 mm laterally. In the presented experiments, itis focused at 80 mm range. Phased arrays send steeredbeams into the imaged medium. One particular beam canbe described by the angle its propagation direction makeswith the normal of the probe. The beams’ directions areequally spaced in angle, spanning an angular width of 90◦.All the array elements are active in receive.

The L12-5 38 mm linear array is a broadband trans-ducer with a 12 to 5 MHz extended operating frequencyrange. It is a 192-element, fine pitch, 38 mm lateral foot-print linear array. In the experiments presented here, it isfocused at a depth of 60 mm. Linear arrays send straightbeams into the imaged medium (the propagation direc-tion of each beam is parallel to the normal of the probe).One particular beam can be described by the lateral (az-imuthal) position of its propagation axis. The field of viewof such arrays is thus laterally limited to the footprint ofthe probe. The number of elements (M) of such arrays canbe greater than the number of signal channels (N) sup-ported by the electronics. The backscattered data is thusrecorded only on a subset of N array elements, which foreach transmit beam are chosen to be symmetrical aboutthe beam’s propagation axis. This receive scheme is re-ferred to as a “walking aperture” scheme because the ac-tive receive aperture “walks” from one end of the array tothe other, following the position of the transmit beam.

Phased array Capon images are obtained exactly as de-scribed in Section II. For linear array images, the sameformulas from Section II are used. The difference is that,for a particular point of interest P , the receive vectors cor-respond to the data that is received only by the N fixedelements that are active in receive for the transmit thathas P on its axis—transmit number k(P ).

For the images presented here, the covariance matrixfor each point is obtained as explained in Section II, withK = 16 observations of each point. It means that 16 trans-


Fig. 2. (a) Phantom DAS image. (b) Phantom Capon image. Theimage brightness is adjusted so that a speckle region in the center ofthe image is displayed at the same brightness in the two images. Thescale is indicated in millimeters. The lateral axis is azimuth, and thevertical axis is depth.

mit beams—ideally laterally sampled at Nyquist and takensymmetrically about the A-line of interest—are necessaryto build one A-line of the final Capon images. However,one transmit beam can be used for several A-lines, and forthe A-lines at the edges of the field-of-view, fewer trans-mits are used to compute the covariance matrix; so thatthe DAS and Capon images are obtained with the sametransmit sequence.

IV. Results

A. Phased Array Phantom Images

Phantom data is first collected with the phased arrayand processed with the DAS and the Capon algorithmsfor comparison (Fig. 2). For display, the brightness of bothimages is adjusted so that a speckle region in the center ofthe image has the same brightness in the two images. Thedisplay dynamic range is the same for both images.

The Capon image exhibits an overall higher contrast, re-duced sidelobes around the point scatterers, and a better

lateral resolution than the DAS image. The reduction insidelobe levels and lateral resolution enhancement is bestappreciated in the near-field of the probe (see the hori-zontal rows of points between 20 and 30 mm depth). Thelateral resolution enhancement and reduction in sidelobelevel also manifest themselves in sharper cyst boundariesand clutter reduction in the cysts on the right-hand sideof the image.

A side effect of the Capon algorithm is to increase thespeckle variance, which is higher in the Capon image thanin the DAS image.

These effects are readily perceived by visual examina-tion of the ultrasound images themselves; they can also bequantified by image metrics. The most significant ones arethat 1) the sidelobe-to-mainlobe ratio around the pointscatterers is 10 to 15 dB lower in the Capon image than inthe DAS image; 2) the lateral size at −6 dB of the pointscatterers in the Capon image is 60 to 80% that of theDAS image (the lateral resolution enhancement is depth-dependent, with more enhancement at low depths and lessenhancement close to the transmit focal depth; 3) the pointscatterer-to-speckle amplitudes ratio and the speckle-to-cyst amplitudes ratio are improved by about 15 dB and10 dB, respectively, from the DAS to the Capon image;and 4) the speckle variance in an area in the center of thelog-compressed images is more than doubled (×2.2) fromthe DAS to the Capon image.

B. Phased Array In Vivo Heart Images

The Capon algorithm is now tested on in vivo phasedarray heart data from a healthy male volunteer (Fig. 3).The brightness of Capon and DAS images is adjusted sothat a speckle area near the center of the field of viewappears at the same brightness in both images. The displaydynamic range is the same for both images.

The smaller speckle size in the near-field of the probeon the Capon image suggests that the Capon image res-olution is better. Most interesting in cardiac imaging, theCapon image also exhibits lower clutter levels in the heartchambers. A few bright points persist within the chambersin the Capon image. Visual inspection of the per-elementsignals corresponding to these locations showed coherentsignals across the aperture, indicating these are real struc-tures as opposed to the consequence of clutter or multipathartifacts. The boundaries between tissue and void are alsobetter delineated in the Capon image. This is seen at themyocardium boundary in the center and the lower-rightpart of the image.

However, as in the phantom image, the Capon algo-rithm increases the speckle variance. This gives the my-ocardium a different texture than in the DAS image.

C. Linear Array Phantom Images

Phantom data is first collected with the linear arrayand processed with the DAS and the Capon algorithms(Fig. 4). For display, each image is normalized to its max-imum pixel value. The display dynamic range is the samefor both images.


Fig. 3. (a) Heart DAS image. (b) Heart Capon image. The imagebrightness is adjusted so that a speckle region within the myocardiumis displayed at the same brightness in the two images. The scale isindicated in millimeters. The lateral axis is azimuth, and the verticalaxis is depth.

The scatterers at 25 mm depth have strong sidelobes inthe DAS image. Indeed, far from the focal depth (60 mmhere), the transmit beam is broad laterally. These sidelobescould be reduced by focusing at a shallower depth.

The Capon image exhibits a better resolution and con-trast than the DAS image. The Capon beamformer is ableto reduce the sidelobes of the point scatterers at depthsshallower than the focus to a point that they are no longervisible in the image. Observing the vertical column of pointscatterers at 10 mm azimuth, one can see that that theCapon algorithm improves the lateral resolution and de-creases the sidelobes level at all depths.

Overall, the contrast of the Capon image is higher. Thisresults in darker speckle, since the images are normalizedto the point scatterers’ pixel values. As in the phased arraycase, the speckle variance is increased in the Capon imagewith respect to the DAS image.

Finally, one can see that the electronic noise that affectsthe lower 10 to 15 mm of the DAS image is reduced by the

Capon beamformer. It makes the hyper-echoic inclusion inthe bottom-left of the image (−10 mm azimuth, 60–70 mmdepth) more visible.

These effects are readily perceived by visual examina-tion of the ultrasound images themselves; they can also bequantified by image metrics. The most significant ones arethat 1) the sidelobe-to-mainlobe ratio around the pointscatterers is about 10 dB higher in the Capon image thanin the DAS image; 2) the lateral size at −6 dB of the pointscatterers in the Capon image is 80 to 90% that of theDAS image (the lateral resolution enhancement is depth-dependent, with better relative enhancement far from thetransmit focal depth); 3) the point scatterer-to-speckle am-plitudes ratio and the speckle-to-cyst amplitudes ratio areimproved by about 10 dB and 5 dB, respectively, from theDAS to the Capon image; 4) the speckle variance in anarea in the center of the log-compressed images is morethan doubled (×2.1) from the DAS to the Capon image;and 5) the level of the electronic noise at the bottom of theimage is decreased by 6 dB from the DAS to the Capon im-age (when taking as a reference the brightness of a specklearea near the center of the image).

D. Linear Array In-Vivo Abdominal Scan

Fig. 5 presents the comparison between the DAS andthe Capon beamformers on an in vivo abdominal scan ofa healthy male volunteer. Each image is normalized to itsmaximum pixel value. The display dynamic range is thesame for both images.

The lateral resolution of the Capon image is greaterthan that of the DAS image. This is illustrated by thepointlike scatterers (vessel walls) in the center of the imageat azimuth 1 mm, depth 9 mm; and azimuth −1 mm, depth18 and 22 mm. These features are more readily detectedin the Capon image than in the DAS image, also becausethe background speckle has lower levels. The reduction ofthe gray levels of speckle in the Capon image gives it agreater apparent contrast than the DAS image. Also, theelectronic noise present deep in the field of view in theDAS image is largely filtered out by the Capon algorithm.

The bright diagonal structures (tissue boundaries) inthe center and right part of the field of view are betterdelineated in the Capon than in the DAS image. However,the continuity of these bright structures is less well repre-sented in the Capon image. This is seen in the horizontallayers in the near-field of array (depth < 5 mm), and ata depth 21 mm, azimuth 15 mm. At these locations, theCapon image shows gaps in bright structures that appearto be continuous in the DAS image. These gaps could re-sult from the Capon beamformer’s tendency to accentuatethe underlying speckle pattern.

V. Discussion

In this section, the advantages of using focused beamsover the synthetic aperture transmit scheme are first re-viewed. The behavior of the present implementation of the


Fig. 4. (a) Phantom DAS image. (b) Phantom Capon image. Each image is normalized to its highest pixel value. The scale is displayed inmillimeters. The lateral axis is azimuth, and the vertical axis is depth.

Fig. 5. (a) Abdominal DAS image. (b) Abdominal Capon image. Each image is normalized to its highest pixel value. The scale is displayedin millimeters. The lateral axis is azimuth, and the vertical axis is depth.


Capon algorithm on real ultrasound data and its interestin medical ultrasound imaging is then discussed. The ro-bustness and computational complexity of the Capon al-gorithm is briefly analyzed.

Using focused beams has several advantages, in ultra-sound imaging in general and for the application of theCapon beamformer in particular. The most obvious ad-vantage is in the signal-to-noise ratio of the data. By us-ing many elements of the imaging array at a time to emitfocused beams, one insonifies the imaged medium with sig-nificantly higher energies than with diverging waves emit-ted by single elements [14]. It results in measured data witha higher signal-to-noise ratio and images with a higher con-trast. Enhancing the signal-to-noise ratio is also criticalwhen it comes to inverting the covariance matrix R(P ).If the signal-to-noise ratio is too low, the lowest (but stillphysically meaningful) eigenvalues of the matrix are lostin the noise and their contribution to the matrix inverseis hidden. The consequence is a reduced efficiency of theweight vector w(P ) to filter out off-axis signals from thereceive vector. Furthermore, the ability to concentrate suf-ficiently high energies on a given point of the imagedmedium can give rise to harmonic generation, enabling theuse of the Capon beamformer with harmonic data.

The second advantage of using focused beams is a re-duced sensitivity to motion, when compared to syntheticaperture images. Indeed, the synthetic aperture imagingscheme uses the data backscattered by point P when in-sonifying the medium with all the transmits to computethe corresponding pixel value. As these transmits have tobe emitted sequentially, the point of interest can move be-tween the first and the last transmit. This blurs the por-tions of the image that are subject to motion. Using fo-cused beams, only one transmit is necessary to computethe pixel value of each point, and this motion artifact is ab-sent. For the same reason, using several transmits (focusedor diverging) as the several observations of point P can in-duce errors in the estimation of the covariance matrix ifthe medium under investigation is moving: scatterers thatare off-axis during one observation can be moved on-axisfor a later observation. Some of the off-axis signal rejectioncapability can be lost because the algorithm sees these sig-nals as on-axis, and on-axis signals can be rejected becausethe algorithm sees them as off-axis. If fewer transmits areused to insonify the medium (as proposed here: 16 trans-mits are used to build the covariance matrix, compared to128 if using a synthetic aperture scheme and a 128-elementarray), the effect of motion is reduced because the mediumunder investigation has less time to move from the first tothe last transmit.

Note that the synthetic aperture-like implementationsof the Capon beamformer can also reduce the sensitivity tomotion by reducing the number of transmits [15], but thisis at the expense of a signal-to-noise ratio that is alreadynot optimal.

The main properties of the Capon images are an in-creased contrast and resolution at all depths and an in-creased speckle variance with respect to the DAS image.

The contrast and resolution enhancements are desirablefeatures. For example, it has been noticed in the clini-cal abdominal scan that point-like scatterers appear moreclearly in the Capon image than in the DAS image. Bylowering the sidelobe level, the Capon algorithm also di-minishes the smearing of bright tissue signals into voidsor less reflective tissue (clutter reduction). This effect isresponsible for a good part of the contrast enhancementof the heart image and the abdominal scan. Having bet-ter defined tissue boundaries also gives more confidence intheir precise location. For example, on the heart image,the lower part of the endocardium is more easily delin-eated in the Capon than in the DAS image. Confidence inthe delineation of the boundary between the myocardiumand the heart chambers is important for accurate segmen-tation of the heart, which is needed for quantification ofcardiac function. For this dataset, we lack a ground truthmeasurement to evaluate which image depicts the bound-ary more accurately. However, the design of the Caponweights, aimed at preserving on-axis signals and reduc-ing off-axis signals, and its validation on phantoms sug-gest that the Capon image is more accurate. Further workwould be required to confirm whether using the Caponalgorithm would improve segmentation quality.

On the other hand, the Capon algorithm tends to in-crease the speckle variance. This has also been pointedout by other investigators using another implementation ofthe Capon beamformer [16]. Indeed, dark bits of speckleappear dark on DAS images because the correspondingper-channel signals are very incoherent across the imag-ing aperture. The Capon algorithm thus tags these sig-nals as off-axis and filters them out, resulting in an evendarker appearance of the dark bits of speckle. This in-crease in speckle variance alters the apparent texture ofthe tissues and might hinder the detection of small darkregions. This effect, however, might be mitigated by theuse of spatial compounding methods, such as SonoCTTM

(Philips Healthcare, Andover, MA) [17].Finally, the fact that the Capon algorithm is able to

filter out the electronic noise at the bottom of the lineararray images is a good thing because it yields higher imagecontrast at large depths. But it is quite a surprise; indeed,the covariance matrix for white noise is the identity matrix,which should yield the same uniform weights as the DASbeamformer. It may be that the electronic noise is whitetemporally but not spatially across the probe aperture;there might thus be some correlation between the noiseof different observations. The Capon beamformer wouldidentify this noise as off-axis signals because it is incoher-ent across the aperture, and filter it out.

All the qualitative effects of the Capon beamformer onthe image appearance have been confirmed by measuringimage metrics such as the PSF parameters and the speckle-to-cyst contrast. However, one has to keep in mind that theCapon beamformer is a nonlinear method. Thus, measur-ing metrics like the parameters of a point-spread functionis no longer very meaningful because the PSF is no longeran intrinsic characteristic of the imaging system.


The most critical step in the Capon algorithm is the ma-trix inversion, which is required for every pixel. The qual-ity of the Capon image strongly depends on this particu-lar step. The approach chosen here is an adaptive matrixinversion. It has been designed in a previous implementa-tion of the Capon beamformer for ultrasound imaging tobe robust to a certain amount of phase aberration [9]. Thematrix inversion algorithm can be tuned to the amount ofexpected phase aberration by adjusting the diagonal load-ing level. We found that the quality of the final Caponimage is quite sensitive to the choice of this parameter.Assuming a weak aberration results in a light diagonalloading of the pseudo-covariance matrix. It can lead tomore artifacts in the image (such as those observed in thefirst abdominal scan; gaps can artificially appear in con-tinuous structures). On the other hand, if too strong anaberration is assumed, it results in heavy diagonal loadingof the pseudo-covariance matrix, yielding images that donot exhibit a significant contrast and resolution enhance-ment with respect to DAS.

The matrix inversion as it is performed here involves asingular value decomposition that, for a matrix of size N ,requires O(N3) operations (N is the number of elementsof the ultrasound array, of the order of one hundred). Thishas to be compared with the computational complexityof the DAS algorithm: To compute one pixel value, onlyN sums are needed—complexity O(N). Significantly moreresources would be needed to implement the Capon al-gorithm in real time. The applications of such adaptivebeamforming algorithms for ultrasound imaging are thuscurrently limited to off-line analysis of previously acquireddata. For example, the Capon image can be used as an in-put to segmentation algorithms that work off-line anyway.

To reduce the computational load of the Capon algo-rithm, further work is needed to try to reduce the size ofthe covariance matrix. One approach would be to dividethe imaging array into M subapertures of N/M elements(for example, four subapertures of 32 elements each in thecase of a 128-elements array), and apply the Capon al-gorithm on per-subapertures beamformed data instead ofper-channel data. Matrices of size 4×4 only would have tobe inverted instead of matrices of size 128 × 128, and thecomputational complexity would be divided by (128/4)3.One could also try to get rid of the matrix inversion stepcompletely, choosing the weight vector w among a finitenumber of pre-defined apodization vectors that satisfy theunity-gain constraint of (7). For each pixel, the one of thesevectors that leads to the lowest value of wR(P )w wouldbe chosen to weight the data.

VI. Conclusions

A new implementation of the Capon beamformer hasbeen presented. This method is designed to work with thestandard acquisition scheme for medical ultrasound data,which involves focused beams rather than diverging ones.

The Capon images exhibit enhanced contrast and res-olution when compared to their DAS counterparts, which

gives them a significantly different overall aspect. The in-creased precision with which tissue boundaries are delin-eated with the Capon algorithm make it a good candidateas an aid to segmentation algorithms.

However, the Capon beamformer is significantly morecomputationally demanding than the traditional DASmethod. This currently limits its application to off-linedata analysis.

References

[1] B. Van Veen and K. M. Buckley, “Beamforming: A versatileapproach to spatial filtering,” IEEE ASSP Mag., vol. 5, no. 2,pp. 4–24, Apr 1988.

[2] J. Capon, “High resolution frequency-wavenumber spectrumanalysis,” Proc. IEEE, vol. 57, no. 8, pp. 1408–1418, Aug. 1969.

[3] J. A. Mann and W. F. Walker, “A constrained adaptive beam-former for medical ultrasound: Initial results,” in Proc. IEEEUltrason. Symp., 2002, pp. 1807–1810.

[4] M. Sasso and C. Cohen-Bacrie, “Medical ultrasound imagingusing the fully adaptive beamformer,” in Proc. IEEE Conf.Acoust., Speech, Signal Process., 2005, pp. 489–492.

[5] J.-F. Synnevag, A. Austeng, and S. Holm, “Adaptive beamform-ing applied to medical ultrasound imaging,” IEEE Trans. Ul-trason., Ferroelect., Freq. Contr., vol. 54, no. 8, pp. 1606–1613,Aug. 2007.

[6] T. J. Shan and T. Kailath, “Adaptive beamforming for coher-ent signal and interference,” IEEE Trans. Acoust. Speech SignalProcessing, vol. ASSP-33, no. 3, pp. 527–536, June 1985.

[7] O. L. Frost, III, “An algorithm for linearly constrained adaptivearray processing,” Proc. IEEE, vol. 60, no. 8, pp. 926–935, 1972.

[8] C. Prada and J. L. Thomas, “Experimental subwavelength lo-calization of scatterers by decomposition of the time reversaloperator interpreted as a covariance matrix,” J. Acoust. Soc.Amer., vol. 114, no. 1, pp. 235–243, July 2003.

[9] Z. Wang, J. Li, and R. Wu, “Time-delay and time-reversal-based robust capon beamformers for ultrasound imaging,” IEEETrans. Med. Imag., vol. 24, no. 10, pp. 1308–1322, Oct. 2005.

[10] J. Li and P. Stoica, Robust Adaptive Beamforming. ch. 1, NewYork: Wiley, 2005.

[11] G. Arfken, “Lagrange multipliers,” in Mathematical Methods forPhysicists. 3rd ed. Orlando, FL: Academic Press, 1985, pp. 945–950.

[12] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge:Cambridge University Press, 1985.

[13] J. Li, P. Stoica, and Z. Wang, “On robust Capon beamformingand diagonal loading,” IEEE Trans. Signal Processing, vol. 51,no. 7, pp. 1702–1715, July 2003.

[14] Y. Wang and M. O’Donnell, “Coded excitation for syntheticaperture ultrasound imaging,” IEEE Trans. Ultrason., Ferro-elect., Freq. Contr., vol. 52, no. 2, pp. 171–176, Feb. 2005.

[15] I. K. Holfort, F. Gran, and J. A. Jensen, “Minimum variancebeamforming for high frame-rate ultrasound imaging,” in Proc.IEEE Ultrason. Symp., 2007, pp. 1541–1544.

[16] J. Synnev̊ag and C. Nilsen, “Speckle statistics in adaptive beam-forming,” in Proc. IEEE Ultrason. Symp., 2007, pp. 1545–1548.

[17] R. Entrekin, P. Jackson, J. R. Jago, and B. A. Porter, “Real-timespatial compound imaging in breast ultrasound: Technology andearly clinical experience,” Medica Mundi, vol. 43, no. 3, pp. 35–43, Sep. 1999.

Francois Vignon graduated from Ecole Nor-male Superieure de Paris in 2002 with a mas-ter’s degree in physics. He received his Ph.D.degree in Laboratoire Ondes et Acoustique,Paris, in 2005. Since 2006, he has been a Se-nior Member of Research Staff at Philips Re-search North America. His main current re-search interests are beamforming and signalprocessing for medical ultrasound, includingtime-reversal techniques.


Michael Burcher earned his M.Eng. de-gree in engineering from Cambridge Univer-sity in 1998. He received his M.Sc. degree inengineering and physical science in medicineand biology from Imperial College, London, in1999. In 2002, he completed his Dr.Phil. de-gree in engineering science at Oxford Univer-sity. Since 2003, he has been a Senior Memberof Research Staff at Philips Research NorthAmerica. In 2007, he was given the J. A.Lodge Award by the Institution of Engineer-ing and Technology for his contributions to

medical engineering. His research interests include beamforming formedical ultrasound, photoacoustics, and signal processing.

capon beamforming in medical ultrasound imaging with focused beams

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