An Adaptive Strain Estimation Algorithm Using Short Term Cross Correlation Kernels and 1.5D Lateral Search

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An Adaptive Strain Estimation Algorithm Using

Short Term Cross Correlation Kernels and 1.5D

Lateral Search

Shaiban Ahmed

Biomedical Engineering

University of Illinois Chicago

Chicago, IL, USA

sahme83@uic.edu

Rasheed Abid

Biomedical engineering

Illinois Institute of Technology

Chicago, IL, USA

r.abid94.bogra@gmail.com

S. Kaisar Alam

Rutgers University

Newark, NJ,USA

kaisar.alam@ieee.org

Abstract

Adaptive stretching, where the post compression signal is iteratively stretched to maximize the correlation between the pre and

post compression rf echo frames, has demonstrated superior performance compared to gradient based methods. At higher levels

of applied strain however, adaptive stretching suffers from decorrelation noise and the image quality deteriorates signiﬁcantly.

Reducing the size of correlation windows have previously showed to enhance the performance in a speckle tracking algorithm

but a correlation ﬁlter was required to prevent peak hopping errors. In this paper, we present a novel strain estimation algorithm

which utilizes an array of overlapping short term cross correlation kernels which are about one-fourth the size of a typical large

kernel, to implement an adaptive stretching algorithm. Our method does not require any supplementary correlation ﬁlter to prevent

false peak errors. Additionally, a lateral search is incorporated using 1.5D algorithm to account for the mechanically induced

lateral shift. To validate the efﬁcacy of our proposed method we analyzed the results using simulation and in-vivo data of breast

tumors. Our proposed method demonstrated a superior performance compared to classical adaptive stretching algorithm in both

qualitative and quantitative assessment. Strain SNRe, CNRe and image resolution are found to improve signiﬁcantly. Lesion’s

shape and boundary are more clearly depicted. The results of improvement are clearly evident at higher levels of applied strain.

Keywords

elastography, strain, stress, ultrasound, compression, B-mode, breast, tumor.

I. INTRODUCTION

Medical imaging is a powerful means of diagnosis of various kind of diseases in the human body [1], [2], [3], [4], [5].

Speciﬁcally, strain Elastography, which has garnered wide scale popularity [6], [7], [8], [9], [10] over the years due to its non-

invasive and inexpensive nature, is primarily an imaging procedure that can map the elastic features of biological tissues and

provide an extensive visual and quantitative analysis of the discriminant tissues properties and can be used to assess intricate

tissue features [11]. As an imaging technique, Elastography has been used in numerous clinical applications such as diagnosis

of breast [1], [12], [13], myocardium imaging [2], renal pathology [3] etc. Pathology and in some cases physiological

phenomena change the stiffness of many tissues and this change can be detected by manual palpation, a method that has

been used for millennia as a diagnostic tool. However, free hand palpation based diagnosis is limited to detection of abnormal

tissues having a signiﬁcant difference in stiffness compared to their surroundings. Also palpation is subjective and clinician

dependent making independent conﬁrmation of ﬁndings difﬁcult. Elastography can recreate palpation like ability by using

advanced computational algorithm to detect and accurately analyze the response from an applied stimuli and thus, examine

tissue features with more efﬁciency and accuracy. Quasi-static strain imaging techniques based on Elastography are compression

based methods where an external excitation or stimulus is generated by mechanically compressing the tissue surface by using

ultrasound transducers [14]. In Strain Elastography, ultrasound echoes are recorded before and after applying the mechanical

compression. These pre and post deformation signals are then used to estimate strain. Numerous correlation based algorithms

have been presented to estimate strain from the pre and post compression signals. These fall into two groups: a) Gradient based

approaches [6], [7], [8], [9], [15], [15] and b) Direct strain estimators [16], [17], [18]. In gradient based methods, strain is

computed by calculating the displacement derivative. Where, displacement due to the applied mechanical compression can be

calculated from time-delay or phase shift. Time delay can be estimated by computing cross-correlation [6], [7], [8], [9] of pre

and post compression signals. Alternatively, phase shift can be estimated from phase domain multiplication [19], [20], [21].

However such methods are prone to noise ampliﬁcation. Stretching the post compression signal using a global stretch factor

[22], [23] prior to correlation computation has previously resulted in noise reduction but only appeared to be effective at low

strain [23]. Median ﬁltering or least square based techniques such as linear regression [24] or smoothing spline [25] can also

be applied to enhance performance. But the applied strain itself induces decorrelation noise which can compound with higher

levels of applied strain. Direct strain imaging techniques have presented more robust performance compared to gradient based

arXiv:2210.12297v1 [q-bio.TO] 21 Oct 2022

methods where the strain is computed directly from pre and post deformation echo signals. Adaptive strain estimation [26]

or an adaptive spectral strain estimation [18] is a direct strain estimation approaching where a local stretch factor is used to

stretch the post compression signal to improve cross correlation accuracy for strain estimation. Adaptive stretching of the post

compression signal instead of global stretching leads to more accurate strain estimates because stretching the post compression

signal using one global stretch factor can lead to inaccurate estimates as the tissue displacement is not homogeneous in all

regions. However, the classical adaptive strain estimation algorithm was implemented using cross correlation windows having

length much longer than the autocorrelation width of the signal. Strain maps generated by such large windows are found to

have low spatial resolution and are prone to decorrelation noise and low SNR at higher levels of applied strain. Shorter cross

correlation windows were used previously [27] and appeared to enhance the performance of a speckle tracking algorithm

where the strain was estimated from a difference estimate but a correlation ﬁlter was required to reduce false peak errors and

to achieve high SNR values. In this paper we investigate the effects of short term cross correlation in a direct strain estimation

approach by implementing adaptive stretching algorithm using short term correlation kernels. Strain is estimated from a mean

short term cross correlation function rather than using an additional correlation ﬁlter to eliminate peak hopping errors. A lateral

search is also conducted to search for the lateral shift and have better strain estimates for high strain values. The performance

of the proposed method has been thoroughly evaluated by using 2-D Finite Element Model simulated data as well as in-vivo

data of breast tumors.

II. METHOD

A. Theory of Ultrasound Signal Model and Adaptive Estimator of Strain:

For the initial analysis a simple one-dimensional (1D) model is developed below to exhibit ultrasound back-scattered radio-

frequency (RF) signals. We can represent the pre and post deformation echo signals by [26]:

r1(t) = s1(t) + n1(t)

or, r1(t) = s(t)∗p(t) + n1(t)

or, r2(t) = s2(t) + n2(t)

or, r2(t) = st

a−to∗p(t) + n2(t)

(1)

Here, r1(t)and r2(t)are the pre and post compression signals respectively; s(t) stands for the effective rf backscatter

distribution function in 1-D model; p(t) is the impulse response of the ultrasonic system; n1(t)and n2(t)are induced

uncorrelated random noise processes and the symbol ‘∗’ represents convolution. For conventional elastography, the applied

strain ”ε” is signiﬁcantly smaller than 1. Hence the constant parameter, ”a” will be close to unity. Mathematically:

a= 1 −ε∼

(2) [when ε1]

The induced displacement between the pre and post compression signals is depth dependent making them jointly non-

stationary. Also, for 1D model, the displacement is assumed to be only in the axial direction. The cross correlation function

between r1(t) and r2(t)computed at time t=tocan be expressed using the equation [27]:

(r1(t)? r2(t)) (τ) = ˆ

R(to, to+τ) = 1

TZto+T

to−

r1(t)r2(t+τ)dt (2)

Here, ”∗”denotes cross correlation between the two signals; τis the induced displacement or correlation lag; T stands

for the correlation window or kernel length. The given time domain functions can be converted to their corresponding spatial

functions (function of their spatial position) by simply using the transformation equation x = tc/2, where x is the distance from

the transducer, c is the propagation speed of sound in the tissue medium [26]. Correlation based techniques are widely used for

estimating strain where the time delay or displacement is computed by locating the position of peak cross correlation co-efﬁcient

[28], [29], [30]. Such correlation based techniques suffer from decorrelation noise which compounds with the increment of

applied strain [23], [29], [30]. In order to reduce the effect of applied strain, stretching the post deformation signal temporally

prior to correlation computation has previously showed to improve the performance of correlation estimates [23], [31]. However,

the biological tissues are Heterogeneous in nature and hence, the strain would vary at different window locality. Thus, a varying

stretch factor would be ideal instead of a global stretch factor that time stretches the entire post compression A-line at one go.

An adaptive algorithm was used previously [26] that uses the stretch factor itself as an estimator of strain. This method uses

an adaptively varying local stretch factor (1/α)to stretch the post compression signal temporally. The stretch factor is modiﬁed

adaptively till the highest cross correlation co-efﬁcient value is achieved. The corresponding value of the stretch factor for

which the peak cross correlation co-efﬁcient is obtained (1/αmax)is then used to estimate strain using the simple relation:

ε= 1 −1

αmax

(3)

Let r3(t)be the stretched version of r2(t)with (1/α)being the stretch factor. Then we can denote this temporally stretched

post compression signal as:

r3(t) = r2(αt) = s3(t) + n3(t) = αs α

at−to∗p(αt) + n3(t)(4)

Now, if we compute the cross correlation between the pre and stretched post compression signal, then the co-efﬁcient of

the cross correlation function can be represented by [26]:

cp13 (to, to+τ) =

TRto+T

to−

r1(t)r3(t+τ)dt

TRto+T

to−

1(t)dt 1

TRto+T

to−

3(t+τ)dt

(5)

cp13 (to, to+τ) =

TRto+T

to−

[s(t)∗p(t)] αs α

a(t+τ)−to∗p(αt)dt

TRto+T

t0−

[s(t)∗p(t)]2dt 1

TRto+T

to−

2αs α

a(t+τ)−to∗p(αt)2dt

(6)

\p13 max =

TRto+T

to−

[s(t)∗p(t)] sα

at∗p(t)dt

TRto+T

to−

[s(t)∗p(t)]2dt 1

TRto+T

to−

2sα

at∗p(t)2dt

≤1(7)

Maximum value of this cross correlation co-efﬁcient can be achieved when α=a. Selecting an appropriate stretch factor

using an adaptive mean, this peak cross-correlation coefﬁcient value is located. Then strain is calculated directly from this

stretch factor using Equation 4. However, this classical adaptive stretching method utilizes large correlation kernels which are

subject to decorrelation noise at higher strain levels. The resultant strain elastograms hence are predicted to have low SNRe

and CNRe values. Also, the large correlation windows can reduce the spatial resolution which in turn is not ideal for imaging

the strain map. This problem can be solved by reducing the length of the correlation windows.

B. Short Term Correlation Kernels:

Error generated due to variance in displacement or time delay estimation can be limited by reducing the length of correlation

kernel. When the length of the correlation kernel becomes smaller, the effect of decorrelation due to applied strain diminishes

signiﬁcantly. However, as the size of these correlation windows become smaller, the normalization terms in equation (6) begin

to ﬂuctuate which increases error variance and may results in peak hopping or appearance of false peak along with the true peak

in the computed cross correlation function. Previous work using shorter kernels, which have showed to improve the performance

in speckle tracking [27] have used a correlation ﬁlter to reduce the effect of false peaks. The correlation ﬁlter multiplies a

number of adjacent correlation function computed at different spatial positions by different weights or ﬁlter co-efﬁcients to

prevent any false peaks. The accurate length of the ﬁlter must be conﬁgured to obtain higher SNRe values. Designing such

correlation ﬁlter can although be computationally intensive. Also, if a number of spatially adjacent correlation functions are

ﬁltered, it can lead to reduced spatial resolution. However, usage of such correlation ﬁlter can be avoided altogether by using

a mean short term cross correlation function as described below. When a number of adjacent short term cross correlation

functions are generated by computing the cross correlation between overlapping short term pre and post compression windows,

false peaks are expected to pop up along with the true peak. But the position of this true peak always remain at the same

location where the false peaks can rise up at different lag positions. Hence, when all the cross correlation functions are summed

together and their mean is computed, the true peak is expected to tower over all the false peaks.

In order to examine this concept, we ran two simulations on a simple 1-D model. The simulations were conducted under

an applied strain of 2%. The outcomes are depicted in Figure 2. The ﬁrst simulation was conducted using conventional large

windows where the cross correlation function was obtained by using a 2.96 mm kernel. The length of the kernel was determined

based on empirical analysis and the length (2.96 mm) that produces the highest cross correlation co-efﬁcient was selected. For

the second simulation, an array of overlapping short term windows were used. Length of each short term windows was selected

to be 0.74 mm based on an empirical analysis where image resolution, strain SNRe and visual inspection was considered. An

array of 13 short term kernels with 75% axial overlapping were computed to cover a length of 2.96 mm to cover the same

spatial distance as the large window. By computing cross correlation between the respective short term kernels and search

windows, a total of 13 short term cross correlation functions were obtained. Then they were added together and their mean was

computed to obtain the mean cross correlation function. The process of generating axially overlapping short term (ST) windows

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AnAdaptiveStrainEstimationAlgorithmUsingShortTermCrossCorrelationKernelsand1.5DLateralSearchShaibanAhmedBiomedicalEngineeringUniversityofIllinoisChicagoChicago,IL,USAsahme83@uic.eduRasheedAbidBiomedicalengineeringIllinoisInstituteofTechnologyChicago,IL,USAr.abid94.bogra@gmail.comS.KaisarAlamRutgersU...

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An Adaptive Strain Estimation Algorithm Using Short Term Cross Correlation Kernels and 1.5D Lateral Search

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