Uniform precision ultrasound strain imaging
G.M. Treece, J.E. Lindop, A.H. Gee and R.W. Prager
Cambridge University Engineering Department
Cambridge CB2 1PZ
Corresponding e-mail: firstname.lastname@example.org
Ultrasound strain imaging is becoming increasingly popular as a way to measure stiffness variation in
soft tissue. Almost all techniques involve the estimation of a field of relative displacements between
measurements of tissue undergoing different deformations. These estimates are often high resolution,
but some form of smoothing is required to increase the precision, either by direct filtering or as
part of the gradient estimation process. Such methods generate uniform resolution images, but
strain quality typically varies considerably within each image, hence a trade-off is necessary between
increasing precision in the low quality regions and reducing resolution in the high quality regions.
We introduce a smoothing technique, developed from the nonparametric regression literature, which
can avoid this trade-off by generating uniform precision images. In such an image, high resolution is
retained in areas of high strain quality but sacrificed for the sake of increased precision in low quality
areas. We contrast the algorithm with other methods on simulated, phantom and clinical data, for
both 2D and 3D strain imaging. We also show how the technique can be efficiently implemented
at real time rates with realistic parameters on modest hardware. Uniform precision nonparametric
regression promises to be a useful tool in ultrasound strain imaging.
Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Deviations from the simple model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Phantom studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Clinical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.43D strain imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5 Conclusions 23
A Matrices 27
B Multigrid implementation27
C Weighted phase variance 28
It seems likely that some form of ultrasonic strain imaging will be adopted into routine clinical
practice, within a decade, to support a still unestablished set of diagnostic tasks, primarily within
the broad category of soft tissue examinations. Applications discussed in the academic literature
have included detection of soft tissue tumours (Garra et al., 1997; Regner et al., 2006; Svensson
and Amiras, 2006), discrimination without biopsy between complex cysts and malignant breast
lesions (Barr, 2006), monitoring of atherosclerosis (de Korte et al., 1998, 2000), detection and grading
of deep vein thrombosis (Emelianov et al., 2002), assessment of skin pathologies (Vogt and Ermert,
2005) and evaluation of myocardial fitness (Kaluzynski et al., 2001).
There are currently a variety of techniques for generating strain images using ultrasound, and
it is not yet clear which of these techniques will be most appropriate for each of these applications.
However, the majority of techniques involve the local estimation of tissue displacement by comparing
radio frequency (RF) ultrasound data acquired at differing tissue deformation states. The tissue
deformation can be induced in a variety of ways: in the remainder of this paper, we will focus on
quasi-static ultrasound strain imaging, where the tissue is deformed by varying the contact pressure
between the probe and the skin surface.However, the algorithms we develop apply equally to
other strain imaging techniques. Many methods have been proposed for displacement estimation,
e.g., (Alam et al., 1998; C´ espedes and Ophir, 1993; C´ espedes et al., 1995; Lindop et al., 2007,
2008e; Lubinski et al., 1999; Maurice and Bertrand, 1999; O’Donnell et al., 1994; Pesavento et al.,
1999; Pinton et al., 2006; Sumi, 1999; Viola and Walker, 2003; Zhu and Hall, 2002). Such methods
typically produce high resolution displacement estimates, however the measurement quality can vary
enormously across a single image, for instance due to variation in signal strength or decorrelation
caused by non-axial movement.
In quasi-static strain imaging, displacement estimation is followed by gradient estimation in
the axial direction. Simple differencing of consecutive samples (Ophir et al., 1991) amplifies the
high-frequency components of the measurement noise. Hence differencing is often achieved by more
complex techniques such as piecewise-linear least squares regression (PLLSR) (Kallel and Ophir,
1997), moving-average filtering (O’Donnell et al., 1994) and staggered strain estimation (Srinivasan
et al., 2002). All such linear techniques can be interpreted as simple differencing followed by filtering
with fixed kernel coefficients. Indeed, we have previously shown that, except in the case where the
entire data set genuinely consists of noisy measurements from a single linear trend (in which case
PLLSR is the optimal filter), simple differencing followed by filtering with a Gaussian-shaped kernel
can achieve lower estimation noise than these methods at the same resolution (Lindop et al., 2008b).
Since both the displacement tracking and filtering techniques make use of kernels with fixed size,
subsequent strain images have fixed resolution but variable quality. However, this variation can be
quantified, since it is straightforward to obtain a reasonable estimate of the precision (inverse of
measurement variance) of each measurement (Lindop et al., 2008a). Strain images require some
form of normalisation to convert the strain into a displayable range, and to reduce variation that is
simply a result of variation in the applied stress (Lindop et al., 2008c). The precision of the displayed
strain value depends both on the displacement estimation precision and on the normalisation value
used at each point in the image. Both of these factors can vary within each image, leading to large
variations in precision which can make strain images hard to interpret.
In order to prevent confusion due to the display of low precision strain data, images are often
suppressed once the overall precision falls below a fixed threshold (Jiang et al., 2007). However,
strain images with low overall precision can still contain high precision regions, and this is exploited
by techniques which combine multiple images, using local strain precision information to ensure the
best data in each image contributes more to the final result. Such data still contains regions of low
precision, but these can be masked by use of a suitable colour wash (Lindop et al., 2008c).
We present here a method for producing strain images with uniform precision and varying reso-
lution, rather than uniform resolution and varying precision. Such images may be easier to interpret:
lack of precision in strain images leads to regions which falsely appear to have strong fine-scale
stiffness variation, whereas lack of resolution leads to high levels of blurring, which is more easily
interpreted. In this case, a colour wash can be additionally used to suppress areas with very low
resolution (rather than low precision as before). Whether this approach is indeed better is clearly
somewhat subjective, hence the results are mostly presented in visual form, so readers can judge for
In Section 2 we describe the principle behind non-uniform smoothing of strain data, followed by
details of our implementation, since computational issues are important in the context of real-time
applications. Section 3 contains an analysis of the resolution and precision of the subsequent strain
estimates, leading to a formulation for uniform output precision. In Section 4, we compare the
technique with PLLSR and Gaussian filtering, including simulations, phantom studies and clinical
examples, for both 2D and 3D strain data. General conclusions are drawn from these results in
To provide a more general framework for smoothing strain images, we follow the roughness penalty
approach to nonparametric regression (NPR) (Green and Silverman, 2004). It should be noted,
however, that the resulting equations can be arrived at from a variety of directions, for instance
variable-kernel smoothing (Silverman, 1984), which lead to different interpretations of largely the
same parameters. Since we are considering regression in the context of image filtering, we want an
approach which allows user control over the extent of filtering (equivalent to the window length in
PLLSR, for example) whilst automating the local smoothing properties. NPR is a good candidate for
this, since it depends on settings which may be thought of as data weights (which can be automatically
chosen for uniform precision) and a smoothing strength (which can be controlled to adjust the level
In principle, since NPR is a linear operation, we could apply it either before differencing the
displacement data, or after, on the strain data. However, there are two key reasons in practice why
it makes more sense to apply it to strain data:
• Strain data varies with the amount of applied stress. Since we really want to visualise stiffness,
strain data needs some form of normalisation before it can be usefully displayed, loosely equiv-
alent to dividing through by an estimate of applied stress. Hence this also results in a change
in data precision: low stress areas are then correctly identified as having lower precision even
though the displacement precision may have been high1. Applying NPR at this stage allows us
to identify such regions correctly, resulting in far better images if the applied stress was highly
non-uniform due to poor probe movement.
• In order to produce a high quality display, strain data is often persisted over a sequence of
images. This persistence must be over normalised strain in order to ensure the image levels
1This is easier to see in the limit of no applied stress — in this case the displacement precision is very high, since
there is no deformation, but the displayed precision must be very low, since it is not possible to measure stiffness if
there was no deformation at all.
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2D strain data on the regression surface are expressed as a vector of length NxNy,
´ sT=?´ s(1,1)
M1and M2are 2(NxNy− Nx− Ny) × NxNy matrices defined such that the vector M´ s lists first
and second differences respectively, throughout the grid of strain data. For example
(2´ s(Nx−1,Ny)−´ s(Nx,Ny)−´ s(Nx−2,Ny))
(2´ s(1,3)−´ s(1,4)−´ s(1,2))
M2´ s =
(2´ s(2,1)−´ s(3,1)−´ s(1,1))
(2´ s(Nx,Ny−1)−´ s(Nx,Ny)−´ s(Nx,Ny−2))
Solving for the optimal regression surface involves the inversion of eq. (4). For C2, the matrix
W +rM2TM2is a key part of this expression. Owing to its sparsity, matrix multiplication is never
applied directly. Simultaneous equations arising from this matrix associated with coordinates (x,y),
away from the edge of the regression surface, have the form
´ s(x+2,y)+´ s(x−2,y)+´ s(x,y+2)+´ s(x,y−2)
= w(x,y)ˆ s(x,y)
´ s(x+1,y)+´ s(x−1,y)+´ s(x,y+1)+´ s(x,y−1)
Multigrid describes a framework for solving inverse problems rather than a black-box solution — the
details of this framework vary with each application. No attempt is made to explain the multigrid
or full multigrid framework here, but sufficient application-specific details are given to implement
multigrid in this case given general knowledge of the framework (Briggs et al., 2000; Press et al.,
Crucial to the technique are restriction and prolongation operators, for transferring the residual
error to and from coarser grids. For NPR using C1, the equations are only second order, and in
this case we can use simple bi-linear interpolation for prolongation, and its adjoint full weighting
for restriction. These are described by stencils. For prolongation, the stencil shows the result in
the fine grid of prolongating a digital impulse in the coarser grid. For restriction, the stencil shows
the weightings applied to the fine grid to generate a data point on the coarse grid. For coarse grids
which are exactly half the resolution of the finer grid, a symmetric restriction stencil should be set
bi-linear interpolation in 2D is:
4of the prolongation stencil for 2D processing,1
8for 3D processing. The prolongation stencil for
CWEIGHTED PHASE VARIANCE
NPR using C2requires the solution of a fourth order equation, and for this we need smoother
prolongation and restriction operators. We base these on the Catmull-Rom spline (Catmull and Rom,
1974), which has the useful property in this case of ensuring, for prolongation, that the second order
difference of intermediate samples is the exact bi-linear interpolation of the second order difference of
the surrounding samples in the coarser grid. In 1D, the intermediate data point is given by applying
the following weightings to the coarser grid data:
The full prolongation stencil for 2D is hence
The same restriction operators were used to restrict the data weights W to each grid level. These
operations can easily be implemented as 1D convolutions in each of the x and y directions.
Multiple V-cycles were used, with Red-black Gauss-Seidel (Varga, 2000) as the smoothing oper-
ator on each grid, except for the coarsest grid (of at least 6 × 6), where a direct solution was found
using band-limited Cholesky decomposition. For NPR using C1, there were 4 iterations of smoothing
at each smoothing stage of the V-cycle. For NPR using C2, with heavy smoothing, a larger number
of smoothing iterations were potentially required at each stage to ensure rounding errors were suf-
ficiently damped. Hence the iteration count was initialised to 4, then increased by 4 whenever the
residual error decreased by less than 1% over a whole V-cycle.
CWeighted phase variance
It was shown by Lindop et al. (2008a) that, under certain simplifying assumptions, the precision p of
the displacement data can be estimated from the complex cross-correlation ρ of matched displacement
windows in the pre- and post-deformation data. For two matched signals r1eiθ1and r2eiθ2
1 − ρ
In practice, this calculation is only valid for reasonably high correlations, and a slight modification
is made to ensure that low correlations lead to sufficiently low precision values
When tracking displacements using Weighted Phase Separation (WPS) (Lindop et al., 2008e),
the windows are matched by directly minimising a weighted sum of phase differences between pre-
and post-deformation data. In this case, the weighted variance of the residual phase differences
σθ between each sample of the matched windows is an alternative measure of the quality of the
displacement estimate. If the phase difference is weighted by the product of the envelope of each
signal r1r2, then
However, this can be related to eq. (13) by assuming that the residual phase differences θ1− θ2
will be small, and re-writing
??r1r2+ i?r1r2(θ1− θ2) −1
ρ ≈ ?
The second term in the numerator of eq. (16) is imaginary, and in any case will by definition be
zero for matched windows, hence
If we also assume that the signal envelopes r1and r2are similar, i.e.
ρ ≈ 1 −σθ
By substituting this into eq. (14) we can now relate precision to the weighted residual phase variance:
This is useful since eq. (19) can be calculated more efficiently than eq. (14) when using WPS,
and is a more direct measure of the residual error in this form of displacement tracking.