Ultrasonics 37 (1999) 511–519www.elsevier.nl/locate/ultras

A multi-scale algorithm for ultrasonic strain reconstructionunder moderate compression

Jing Bai *, Chuxiong Ding, Yu FanDepartment of Electrical Engineering, The School of Life Science and Engineering, Tsinghua University, Beijing, People’s Republic of China

Received 11 February 1999; received in revised form 7 July 1999

Abstract

Since the waveform of an echo is more distorted under larger compression, elastography can be applied in cases where thecompression is only a few percent. To reduce errors due to distortion of the echo waveform, a novel algorithm (i.e. the multi-scale correlation algorithm) for strain profile reconstruction is proposed in this article. The basic idea of this method is to simulatehuman vision: first, we locate the region of the target, and then match the detail within this region and make a comprehensivecriterion. Based on this concept, three different approach algorithms are proposed and investigated: (1) the two-step method, (2)the multi-scale correlation algorithm, and (3) the extended multi-scale correlation algorithm. To evaluate the algorithms proposedin this work, a computer simulation model is used and a sequence of simulation experiments are performed. The correlatingspecificity is used as a criterion for evaluating the performance characteristics of the correlation estimation. The results show thatthe extended multi-scale correlation algorithm performs significantly better than the other two methods proposed in this articleunder moderate compression ratios. The extended multi-scale correlation algorithm is applicable to both small and moderatecompression ratios. This novel algorithm may provide a useful tool for the clinical application of elastograms. © 1999 ElsevierScience B.V. All rights reserved.

Keywords: Compression; Correlation; Elastography; Multi-scale; Simulation; Strain reconstruction; Ultrasound

1. Introduction then be computed based on this estimation of the scatterdisplacement for the region of interest. It has been

Although we cannot hear ultrasound, ultrasonic reported that the length and the overlap of the cross-imaging helps us to see through the human body. Also, correlation window used are important factors for thethe recently developed elastography technique also has reconstruction of the strain profile. There is an optimalthe potential to help us to feel the hardness of soft window length for obtaining the minimized strain esti-tissues and tumors with ultrasonic echoes. Elastography mation variance [4,7,8]. In our previous studies [9–11],is a method by which the elastic behavior of soft tissues the echo segment within the correlation window wascan be imaged. This method was proposed by Ophir called the tracing segment, and a novel factor, namelyet al. in 1991 [1] and has been investigated by many the correlating specificity (CS), was introduced to evalu-workers since then [2–6 ]. Using differential displace- ate the optimal window length and the ultrasonic systemments of the tissue elements due to quasi-static tissue parameters for elastography. Another error source incompression, a gray-scale axial strain image can be elastography is the distortion of the echo waveformsobtained, which is called an elastogram [1]. The main due to compression [7,11]. Since the echo waveform isprinciple used in constructing elastograms is a correla- more distorted under larger compression, the applicationtion analysis of echoes pre- and post-compression [1]. of elastography is limited to cases where the compressionThe positions of specified echo segments pre- and post- is only a few percent. For such small compressions, thecompression can provide information about scatter signal-to-noise ratio is very low. Thus, the echo wave-movement due to compression. The strain profile can form distortion becomes a big barrier for the clinical

application of the elastography technique. Many efforts* Corresponding author. Tel.: +86-10-6278 4296; fax: +86-10-6278

have been made to overcome this barrier, such as the3057.E-mail address: deabj@tsinghua.edu.cn (J. Bai) temporal stretching method [7], the adaptive strain

0041-624X/99/$ – see front matter © 1999 Elsevier Science B.V. All rights reserved.PII: S0041-624X ( 99 ) 00026-8

512 J. Bai et al. / Ultrasonics 37 (1999) 511–519

estimator [12], strain estimation using the echo envelope from the same corresponding tissue segment (i.e. tracingsegments) should maximize the correlation coefficientmethod [13], and tracing echo segment selection algo-

rithms [11]. To further reduce the errors due to echo without waveform distortion. This is used as an impor-tant criterion for recognition of the tracing segment inwaveform distortion, a novel algorithm (i.e. the multi-

scale correlation algorithm) for strain profile reconstruc- the compressed echoes. Under moderate compression,the specificity of the tracing segment may be distorted,tion is proposed in this article.

In this paper, we first present the background for the which results in estimation errors. It is observed thatthe window length used in the calculation of the cross-development of the multi-scale strain reconstruction

algorithm. Then, the multi-scale algorithm is proposed correlation influences the strain estimation error. Tocharacterize and quantify the specificity of the correla-and described. To validate the proposed algorithm, a

computer simulation model is used and simulation tion function on recognizing the tracing segment, aparameter called the correlating specificity (CS) is intro-experiments are performed. Then, simulation results are

presented and summarized. duced and defined as follows [10]:

2. Background CS=

peak value of cross-correlation

of the tracing segments

0.2 ∑ other five peak values

of the cross-correlation

. (3)

The basic principle of elastography is Hooke’s law.An elastic material under static compression is

With the above definition, CS represents the distinct-deformed, and this deformation is proportional to theness of the tracing segments using the cross-correlationstress applied. Thus, the strain profile will reflect theestimation. If CS is close to unity, it is impossible toelastic properties of the material under investigation. Asdetermine the tracing segment using the cross-correlationproposed by Ophir et al. [1], by using ultrasonic echoesmethod. The results of our previous studies indicatepre- and post-compression of soft tissues, the strainthat, in the sense of maximizing the CS, there is anprofile can be obtained by using the cross-correlationoptimal window length corresponding to a compressiontechnique to estimate the time shift between congruentratio, and CS is close to unity for compression ratiossegments in an A-line pairs as follows:greater than 2% [10]. This is in agreement with theobservation made in Ref. [13]. In this paper, we use thiss(i )=

Dt(i )−Dt(i−1)

DT, (1)

correlating specificity (CS) as a criterion to evaluate thealgorithm proposed.

where s(i) is the average strain of the ith and (i−1)thecho segments, Dt(i) is the estimated time shift of theith segment pair pre- and post-compression, and DT is

3. Methodsthe window length applied.The time shift Dt(i) is determined by locating the

To overcome the limitation on large compression, amaximal peak of the cross-correlation function pre- andmulti-scale estimation algorithm is proposed in thispost-compression echoes. If x represents the ultrasonicwork. Due to compression, cross-correlation (or wave-echoes pre-compression and y represents the ultrasonicform matching) is a difficult task. Good matching ofechoes post-compression, the cross-correlation C(k) canone part will cause a worse matching of the other part,be computed byas shown by Fig. 1. The basic idea of our method is tosimulate human vision: first, we locate the region of thetarget, and then we match the detail within this region.C(k)=

∑i=1L0 x(i )y(k+i )

S∑i=1L0 x2(i ) ∑

i=1L0 y2(k+i )

, (2)Based on this concept, three different algorithms areproposed and investigated: (1) the two-step method, (2)the multi-scale correlation algorithm, and (3) theextended multi-scale correlation algorithm.where L0 is the sampling number within the window

length DT.As reported in Ref. [13], such a cross-correlation 3.1. Two-step method

based strain estimation is only applicable for fairly smallcompression ratios (within 1–2%). In the case of moder- This method consists of adding a refined matching

procedure to the normal cross-correlation based strainate compression, distortion of the echo waveform willcause serious errors in the strain estimation. estimation scheme. As illustrated in Fig. 2, the optimal

window length is used in the normal cross-correlationTo perform the correlation calculation, the ultrasonicechoes are windowed into segments. The segment pair calculation and tracing segment location. The compres-

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Fig. 1. Waveform mismatch caused by compression.

Fig. 2. Estimation scheme for the two-step method.

sion of the tracing tissue segment is calculated based on shift in the refined matching procedure, the location ofthe tracing segment will be modified as (d+Dd ).the maximal correlation location. To compensate for

the mismatch of the pre- and post-compression wave-forms, a refined matching procedure is applied after thenormal estimation process. This refined matching pro- 3.2. Multi-scale correlation algorithmcedure is described in Fig. 2. A small segment in thecenter of the tracing segment is taken as a refined tracing Moderate compression will cause distortion of the

waveform and may lead to inaccurate correlating effectssegment, and the cross-correlation estimation is appliedto this refined matching procedure to determine the between the pre- and post-compression echoes. To be

able to apply cross-correlation based elastography tomodification for the normal matching location. If drepresents the location of the maximal cross-correlation the case of moderate compression, the correlation func-

tion should be modified in order to reduce waveformin the normal estimation and Dd represents the matching

514 J. Bai et al. / Ultrasonics 37 (1999) 511–519

Fig. 3. Illustration of subwindow length scale and subsegmentation for the multi-scale correlation algorithm

distortion effects. Thus, a multi-scale correlation algo- sponding to the maximal CS is taken as the windowrithm is proposed. length for the zeroth scale level. The window length for

Instead of only computing the cross-correlation in the first scale level is 1/n of the zeroth level, the lengthone fixed window length, a comprehensive cross-correla- for the second level is 1/n of the first level, and so on.tion coefficient C∞(k) is defined over multiple window Thus, the comprehensive correlation coefficient C∞(k) islengths, i.e. an averaged superposition of the cross-correlation over

multi-scale window lengths which represents the corre-lating characteristics of the tracing segment not only atC∞(k)=

1

N AC0+∑

i=1m

∑j=1ni

CijB, (4)

the optimal window length scale but also at manyfine scales.where C0 is defined by Eq. (2) for the window length

of L0, m is the total number of subscales involved, n isthe number of subsegmentations for the next scale level,

3.3. Extended multi-scale correlation algorithmN is the number of terms for the summation, and Cij

isthe maximal cross-correlation coefficient for the jth

The result for the multi-scale correlation algorithm issubsegment at the ith scale level, i.e.encouraging, but the computation is elaborate. In fact,upon increasing the scales, the effect of the lower levelcoefficients is reduced. Therefore, an extended multi-C

ij(k)= max

0≤t≤l C ∑n=1Li x

ij(n)y(k+t+n)

S ∑n=1Li x2

ij(n) ∑

n=1Li y2(k+t+n)D, (5)

scale correlation algorithm is proposed as follows.Instead of going through all the scale levels, the

extended algorithm goes directly to the finest scale andwhere L

iis the sampling number within the subsegmen- calculates the comprehensive correlation coefficient only

tation length at the ith scale level, and l is taken to be at this finest level. Thus, the comprehensive cross-half the wavelength. correlation coefficient C◊(k) for the extended multi-scale

The subwindow length scale and subsegmentation areillustrated by Fig. 3. The optimal window length corre-

Fig. 5. Correlation specificity for the extended correlating algorithmFig. 4. Correlating specificity for the multi-scale correlation algorithm. under compression ratios of (a) 0.5%, (b) 4.5%, (c) 8.5% and (d) 12.5%.

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Fig. 6. Reconstructed strain profiles with the single-step method underFig. 7. Reconstructed strain profiles with the two-step method undercompression ratios of (a) 1% and (b) 2.5%.compression ratios of (a) 1% and (b) 2.5%.

correlation algorithm is defined as follows:

coefficient for the extended multi-scale correlation algo-rithm, it is clear that this correlation coefficient is an

C◊(k)=G 0 C0(k)≤0

1

N∑n=1N

Cn(k) C

0(k)≥0H (6) average over the fine scale subsegmentation. The wave-

forms are matched at the fine scale level. The matchingresults are superposed to form a comprehensive correla-tion function for strain estimation. The correlationwhere C

nis the maximal cross-correlation coefficient for

calculation at the zeroth level is only for screening. Onlythe nth subsegment within the window L0, N is thethose segments with non-negative correlations will benumber of subsegments taken for this window length,forwarded into the further fine segmentation and theand C0 is defined in Eq. (2). C

nis calculated by following

comprehensive correlation calculation.formula:

4. ResultsCn(k)=max

t C ∑i=1L1 x

n(i )y(k+t+i )

S∑i=1L1 x2

n(i ) ∑

i=1L1 y2(k+t+i )D, (7)

To evaluate the algorithms proposed in this work, acomputer simulation model is used and a sequence ofsimulation experiments are performed. A detailedwhere L

1is the sampling number within the subsegmen-

tation length and t is the small neighbor region for description of the simulation model can be found in ourprevious works [9,10]. However, a brief description issearching the maximum correlation coefficient.

From the definition of the comprehensive correlation given here for completeness. The inspected medium is

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Fig. 8. Reconstructed strain profiles with the multi-scale correlation algorithm under compression ratios of (a) 1%, (b) 2.5% and (c) 6%.

simulated with an elasticity profile with distributed tion. A few examples of our simulation results arepresented below.scatters inside. The scattering function consists of uni-

formly distributed point scatters with a density of 50 Fig. 4 represents the effects on the correlating speci-ficity for the multi-scale correlation algorithm. For com-scatters per cm. The scatter diameters have a Gaussian

distribution with an average of 0.05 mm, a standard parison, the CS curve for a single scale is also plottedon in Fig. 4 (dashed line). In this case, only two scalesdeviation of 0.01 mm, a maximum of 0.1 mm and a

minimum of 0.01 mm. The interrogated waveform is are used and the subsegmentation number is n=4. Thecompression ratio is 1.5%. This result indicates that byconvoluted with the scattering function to obtain the

simulated echoes. To simplify the problem, the stress adapting the multi-scale algorithm, the overall correla-tion specificity is increased and a wide range of windowprofile and the elastic modulus profile are assumed to

be uniform. The ultrasonic velocity within the medium length can be used.Fig. 5 represents the effects of the subsegmentationis assumed to be constant at 1540 m/s. The sampling

rate is taken as 50 MHz. number on the correlating specificity for the extendedmulti-scale correlation algorithm. To study the correla-In the simulation experiments, the following aspects

are investigated: (1) the correlating specificity (CS) for tion properties under different compression ratios, thefour curves in Fig. 5 represent the correlation specificitythe multi-scale correlation algorithm, (2) the correlating

specificity (CS) for the extended multi-scale correlation under four compression ratios (i.e. 0.5%, 4.5%, 8.5%and 12.5%). These results suggest that increasing thealgorithm, (3) the effect of the subsegment number, (4)

the effect of the subsegment length, and (5) a comparison subsegment number can improve the correlation speci-ficity, and that increasing the compression ratio willof the three algorithms with regard to strain reconstruc-

517J. Bai et al. / Ultrasonics 37 (1999) 511–519

Fig. 9. Reconstructed strain profiles with the extended multi-scale correlation algorithm under compression ratios of (a) 1%, (b) 2.5%, (c) 6% and(d) 9%.

reduce the correlation specificity. This result also shows pression ratio is increased to 2.5%, the single-step andtwo-step methods fail to give the correct reconstruction,that, by adapting the extended multi-scale correlation

algorithm, recognition of the tracing segment is still while the results from the third method is still good.For 6% and 9% compression, the extended multi-scalepossible by calculating the comprehensive correlation

coefficient with more subsegments for moderate com- correlation algorithm is the only valid algorithm ofthe three.pressions (i.e. >10%).

Figs. 6–9 present the strain reconstruction resultsusing the three algorithms proposed in this paper. Forcomparison, Fig. 6 presents the reconstructed strain 5. Discussion and conclusionprofile based on the single-scale algorithm. In Figs. 6–9, the strain profiles are plotted using dotted lines and In this article, we have proposed a novel strain

estimation algorithm for elastograms. We have demon-the reconstructed strain profiles are plotted using solidlines. Fig. 7 presents the results obtained using the two- strated that the extended multi-scale correlation algo-

rithm performs significantly better than the other twostep method, Fig. 8 shows the results obtained using themulti-scale correlation algorithm, and Fig. 9 shows the methods proposed in this article, at least for moderate

compression ratios. The correlating specificity is used asresults obtained using the extended multi-scale correla-tion algorithm. Comparing the results shown in Figs. 6– a criterion for evaluating the performance characteristics

of the correlation estimations.9, we can see that in the case of 1% compression, thereis no significant difference in the strain reconstructions The two-step method is merely a modification on the

conventional correlation method at the peak locationobtained using different the algorithms. When the com-

518 J. Bai et al. / Ultrasonics 37 (1999) 511–519

Fig. 10. Illustration of optimal matching with subsegments.

point. By comparing Figs. 6(a) and 7(a), we can see will introduce more errors into the correlation for theupper level, as shown in Fig. 1. This error will reducethat the two-step method is able to reduce the strain

reconstruction error, but unable to save the correlation the specificity of the correlation estimation. With onlya piecewise correlation for the finest scale, as shown infrom waveform distortion. Thus, as shown in this work,

it does not work well for moderate compression. Fig. 10, only the optimal match is counted, which yieldsan optimal estimation of correlating property while theThe multi-scale algorithm simulates human vision. It

first determines the region of interest and then makes waveform distortion effect is eliminated.In conclusion, the extended multi-scale correlationfine matches. Breaking the correlation routine of match-

ing pre- and post-compression echoes, the post-compres- algorithm is applicable to both small and moderatecompression ratios. The comprehensive cross-correla-sion piece is subsegmented into fine pieces and the fine

pieces are matched with the pre-compression echoes tion coefficient compensates waveform mis-match dueto distortion. This novel algorithm may be a useful toolwithout the constraint of the other pieces. Since optimal

matching is carried on a small neighborhood, confusion in the clinical application of elastograms.of the subwaveforms can be avoided. Therefore, theprinciple of comprehensive correlation calculation is tocut the correlation calculation into multiple subsegment

Acknowledgementmaximal correlating calculations. By cutting the com-pressed echo into pieces and making the best match, the

This work was supported in part by the Chinesemismatching problem shown in Fig. 1 is overcome andNational Natural Science Foundation.better matching is achieved. This is illustrated by Fig. 10,

where the solid line represents the echo pre-compressionand the dashed line represents the echo post-compres-sion, and the latter is cut into pieces for optimal

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