DSpace at VNU: IMAGING ULTRASONIC DISPERSIVE GUIDED WAVE ENERGY IN LONG BONES USING LINEAR RADON TRANSFORM

Ultrasound in Med. & Biol., Vol. 40, No. 11, pp. 2715–2727, 2014
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Original Contribution
IMAGING ULTRASONIC DISPERSIVE GUIDED WAVE ENERGY IN LONG BONES
USING LINEAR RADON TRANSFORM
THO N. H. T. TRAN,* KIM-CUONG T. NGUYEN,*y MAURICIO D. SACCHI,z and LAWRENCE H. LE*zx
* Department of Radiology and Diagnostic Imaging, University of Alberta, Edmonton, Alberta, Canada; y Department of
Biomedical Engineering, Ho Chi Minh city University of Technology, Ho Chi Minh city, Vietnam; z Department of Physics,
University of Alberta, Edmonton, Alberta, Canada; and x Department of Biomedical Engineering, University of Alberta,
Edmonton, Alberta, Canada
(Received 24 October 2013; revised 21 May 2014; in final form 23 May 2014)

Abstract—Multichannel analysis of dispersive ultrasonic energy requires a reliable mapping of the data from the
time–distance (t–x) domain to the frequency–wavenumber (f–k) or frequency–phase velocity (f–c) domain. The
mapping is usually performed with the classic 2-D Fourier transform (FT) with a subsequent substitution and interpolation via c 5 2pf/k. The extracted dispersion trajectories of the guided modes lack the resolution in the transformed plane to discriminate wave modes. The resolving power associated with the FT is closely linked to the
aperture of the recorded data. Here, we present a linear Radon transform (RT) to image the dispersive energies
of the recorded ultrasound wave fields. The RT is posed as an inverse problem, which allows implementation of
the regularization strategy to enhance the focusing power. We choose a Cauchy regularization for the highresolution RT. Three forms of Radon transform: adjoint, damped least-squares, and high-resolution are described,
and are compared with respect to robustness using simulated and cervine bone data. The RT also depends on the
data aperture, but not as severely as does the FT. With the RT, the resolution of the dispersion panel could be
improved up to around 300% over that of the FT. Among the Radon solutions, the high-resolution RT delineated
the guided wave energy with much better imaging resolution (at least 110%) than the other two forms. The Radon
operator can also accommodate unevenly spaced records. The results of the study suggest that the high-resolution
RT is a valuable imaging tool to extract dispersive guided wave energies under limited aperture. (E-mail: lawrence.
) Ó 2014 World Federation for Ultrasound in Medicine & Biology.

Key Words: Ultrasound, Cortical bone, Axial transmission, Guided waves, Dispersion, Phase velocity, Fourier
transform, Radon transform, Aperture, Spectral resolution.

the structure. These waves are generated by the interaction
of elastic waves (compressional [P-waves] and shear [Swaves]) with the boundaries. For guided waves within a
plate, waves are multiply reflected at the boundaries with
mode conversions, that is, P / S or S / P. The boundaries facilitate multiple reflections and also guide the
wave propagation; the waveguide also retains the guided
wave energy and keeps it from being spread out, thus allowing the guided waves to travel long distances within
the plate (Lowe 2002). The plate vibrates in different vibration modes, which are known as guided modes.
Guided modes are dispersive and travel with velocities that vary with frequency. The velocity of a guided
mode depends on material properties, thickness, and frequency. The dispersion curve, which describes their relationship, is fundamental to guided wave analysis. The
dispersion curve can be obtained by finding a solution to
the homogeneous elastodynamic wave equation (Rose

INTRODUCTION
Ultrasonic guided waves have seen many successful industrial applications in non-destructive evaluation and inspection. Guided wave testing technologies have been applied
to material inspection, flaw detection, material characterization, and structural health monitoring (Rose 2004).
Also popular are surface wave methods (Cawley et al.
2003; Masserey et al. 2006; Temsamani et al. 2002;
Tsuji et al. 2012) that characterize near-surface materials
in shallow geologic prospects, structural engineering,
and environmental studies. Surface or guided waves
require a boundary or structure for their existence. Their
propagation is constrained to the near surface or within

Address correspondence to: Lawrence H. Le, Department of
Radiology and Diagnostic Imaging, University of Alberta, Edmonton,
Alberta, Canada T6G 2B7. E-mail:
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1999). The displacement vectors, N, are first assumed general forms with unknown constants. This leads to a set of
equations for the unknowns in matrix form, M 3 N 5 0,
where M is the coefficient matrix of elastic constants,
densities, thickness of the structure, wavenumber, and
frequency. The dispersion or characteristic equation of
guided modes is obtained by setting the determinant of
M equal to zero, that is, jMðw; kÞj 5 0 where u is the
angular frequency and k is the wavenumber. The characteristic equation is non-linear, and numerical solutions
are usually sought.
In recent years, quantitative ultrasound has been
used to characterize material properties of long bones
in vitro (Camus et al. 2000; Le et al. 2010; Lee and
Yoon 2004; Lefebvre et al. 2002; Li et al. 2013; Ta et al.
2009; Tran et al. 2013a; Zheng et al. 2007). The axial
transmission technique is the most common method
used to study long bones. The measurement places the
transmitter and receiver on the same side of the bone
sample. Usually two transducers are employed, where
one transducer is a stationary transmitter and the other
transducer is moved away from the transmitter at a
regular spacing interval to receive the signal. Ultrasound
acquisition systems with one array probe (Minonzio
et al. 2010; Nguyen et al. 2013a) and two array probes
(Nguyen et al. 2014) have also been used. The acquisition

configuration has been applied successfully by Le et al.
(2010) to analyze bulk waves arriving at close source–
receiver distances. Quantitative guided wave ultrasonography (QGWU) is particularly attractive because of the
sensitivity of guided waves to the geometric, architectural, and material properties of the cortex. The cortex
of long bone is a hard tissue layer bounded above and
below by soft tissue and marrow, resulting in highimpedance contrast interfaces, and therefore is a natural
waveguide for ultrasonic energy to propagate. Albeit the
studies using guided waves are limited, the results so far
suggest the potential use of QGWU to diagnose osteoporosis and cortical thinning. The use of ultrasound to characterize bone tissues and evaluate bone strength has
gained some success. A recent publication provides
some updates on experimental, numerical, and theoretical
results on the topics (Laugier and Haiat 2011).
Multichannel dispersive energy analysis requires
reliable mapping of the ultrasound data from the 2-D
time–distance (t–x) space to the frequency–wavenumber
(f–k) space. The mapping is usually performed by the 2-D
fast Fourier transform (2-D FFT) (Alleyne and Cawley
1991). The frequency–phase velocity (f–c) space can later
be obtained by substitution and interpolation via c 5 u/k.
Two-dimensional FFT-based spectral analysis has been
used to study dispersive energies of guided waves propagating along the long bones; however, the extracted
dispersion curves lack the resolution in the transformed

Volume 40, Number 11, 2014

space (f–k or f–c) to discriminate wave modes. The
resolving power associated with the 2-D FFT is linked
to the limited aperture of the recorded data. Because of
the limited aperture, the energy information is spread or
smeared, which makes identification of the dispersive

modes difficult. In clinical studies, the spatial aperture
is limited by the accessibility of the adequate skeleton
length, regularity of the measuring surface, length of
the ultrasound probe and number of channels. Several
techniques have been attempted with some success to
improve the resolution of the dispersion curves, such as
using 2-D FFT in combination with an autoregressive
model (Ta et al. 2006b), group velocity filtering
(Moilanen et al. 2006), and singular value decomposition
(Minonzio et al. 2010).
The RT owes its name to the Austrian mathematician
Johann Radon (1917) and is an integral transform along
straight lines, which is known as a slant stack in
geophysics. The inverse RT is widely used in tomographic
reconstruction problems, where images are reconstructed
from straight-line projections such as x-ray computed assisted tomography (Herman 1980; Louis 1992). The RT
has rarely been used to process ultrasound data. Most
recently, the RT was used to perform ultrasonic Doppler
vector tomography to reconstruct blood flow distribution
(Jansson et al. 1997) and to detect linelike bone surface
orientations in ultrasound images (Hacihaliloglu et al.
2011).
McMechan and Yedlin (1981) generated the first
phase velocity dispersion curves based on the RT of the
seismic wave fields. The data were first slant-stacked
(Radon transformed) to the slowness–intercept (p–t)
domain, which was then followed by a Fourier transform
into the slowness–frequency (p–f) plane. However, the
extracted energies were significantly smeared, and the
dispersion trajectories had poor resolution. The lowresolution dispersion map showed the neighboring modes

clustered together, making modal identification a difficult
task. Over a decade, various computational strategies
(see, e.g., Trad et al. [2002] and Sacchi [1997]) have
been developed in the geophysics community to improve
Radon solutions with enhanced resolution. Recently,
Luo et al. (2008a, 2008b) successfully used the highresolution Radon solution developed by Trad et al.
(2002) to image dispersive Rayleigh wave energies in
geophysical surface wave data. Nguyen et al. (2014)
applied an adjoint RT to study guided wave dispersion
in brass and bone plates.
In this work, we apply the linear RT to extract
dispersive information from ultrasound long bone data.
We present the background theory and three solutions
of the linear Radon transform: standard or adjoint Radon
transform (ART), damped least-squares Radon transform
(LSRT), and high-resolution Radon transform (HRRT).


Imaging in long bones using linear Radon transform d T. N. H. T. TRAN et al.

We compare the resolution of the RT solutions and the FT
solution using a dispersive wave-train data set. Finally
we use the RT to image the dispersion curves from the recorded ultrasound wave fields from a cervine long bone.
To our knowledge, our group is the first to use RT to
analyze ultrasound wave fields propagating in long bones
(Le et al. 2013; Tran et al. 2013b, 2013c; Nguyen et al.
2013b, 2014). Our work is novel in that this article
reports our experiments in which the HRRT was used in
the ultrasound bone study. We indicate the advantages
and robustness of the RT with respect to the following:

the RT does not require regular channel spacing; it can
handle missing records; it requires a smaller aperture of
the recorded data; the HRRT has much better resolving
power over the conventional FT and other RT solutions.
Linear Radon transform
Let d (t,xn ) be a matrix of the multichannel ultrasound time records acquired at offsets (source–receiver
distances), x0 , x1 , ., xN21 , where t denotes the traveling
time and the receivers’ spacing, Dx, is not necessarily
uniform. The discrete linear RT, also known as the t–p
transform, is defined by summing the amplitudes along
a line t 5 t 1 px with move-out px where p is the ray
parameter (or slowness) and t is the zero-offset time
intercept (Ulrych and Sacchi 2005). We write the time
signals, d, as a superposition of Radon signals, m(t, p):
dðt; xn Þ 5

K 21
X

mðt 5 t2pk xn ; pk Þ;

n 5 0; .; N21 (1)

k50

where the ray parameter is sampled at p0 , p1 , ., pK21 .
Taking the temporal Fourier transform of (1) yields
Dðf ; xn Þ 5

K 21

X

Mðf ; pk Þe2i2pfpk xn

(2)

k50

where f is the frequency. In matrix notation, eqn (2)
becomes
D 5 LM
where L is the linear Radon operator
2 2iup x
3
e 0 0 / e2iupK21 x0
5
L54
«
1
«
2iup0 xN21
2iupK21 xN21
e
/ e

(3)

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where LH is the adjoint or complex-conjugate transpose

operator. The adjoint operator is a matrix transpose and
is not the inverse operator. The transformation by LH is
not unitary, that is, LLH s I. However, the adjoint sometimes outperforms the inverse operator in the presence of
noise and incomplete data information (Claerbout 2004).
Nguyen et al. (2014) used the adjoint operator to study
dispersive energies in brass and bone plates. The ART
suffers localization problems, has poor resolution, and
smears the dispersion (Ulrych and Sacchi 2005).
Damped least-squares Radon transform
In real life, the data contain noise, N:
D 5 LM1N:

(6)

We seek a Radon solution or model, M, which minimizes
the following cost or objective function in a least-squares
sense:
J 5 kLM2Dk 1mkMk :
2

2

(7)

The first term is the misfit term, which measures how well
the model predicts the data, and the second term is the
regularization term. The regularization refers to the
constraint imposed explicitly on the estimated model during inversion. The purposes of the regularization term are
to improve the focusing power of the solution and to
stabilize the solution. The degree of contribution of the

regularization term depends on the value of the tradeoff parameter or hyper-parameter, m. In this case, the regularization term is the quadratic length of the model. By
taking the derivative of J with respect to the model M and
equating to zero, we obtain the damped least-squares
solution (Menke 1984)
21

M DLS 5 ðLH L1mIÞ LH D:

(8)

High-resolution Radon transform
We also consider a non-quadratic regularization based
on a Cauchy distribution (Sacchi 1997). The Cauchy probability distribution function induces a sparse model and
minimizes side lobes of the spectra, thus rendering highresolution focusing. The corresponding cost function is

(4)
J 5 kLM2Dk 1m
2

K 21
X
À
 Á
ln 11Mk2 s2

(9)

k50

with u 5 2pf.

Adjoint Radon transform
A simple or low-resolution solution, M, can be
calculated using the equation

where Mk is the slowness-spectral scalar at pk , that is,
Mk 5 Mðf ; pk Þ, and s2 is the scale factor of the Cauchy
distribution. By minimizing the cost function, we arrive
at the high-resolution Radon solution

M Adj 5 LH D

M HR 5 ðLH L1mQðMÞÞ LH D

(5)

21

(10)


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where QðMÞ is a diagonal weighting matrix, that is,
1
Qkk 5 À
2 Á :

11ðMk Þ s2

(11)

Equation (10) is a non-linear system of equations and
can be solved iteratively with the IRLS (iteratively reweighted least-squares) scheme for each frequency
(Scales et al. 1988). On the basis of our experience, four
iterations are sufficient to obtain a reasonably good result.
The Appendix provides further details of the IRLS method
and an implementation of the HRRT algorithm.
METHODS
Simulation
We simulate a linear dispersive wave train with the
spectrum
Â
Sðf Þ 5 Wðf Þe

2i2pf

À

x 2t
cðf Þ 0

ÁÃ
(12)

where Wðf Þ is the spectrum of the source wavelet
and t0 is a time constant. The phase velocity, c (f), is
described by

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ffi
cðf Þ 5 cmin 1ðcmax 2cmin Þ
(13)
11ðf =fc Þ
where cmin is the minimum phase velocity, cmax is the
maximum phase velocity, and fc is the critical frequency.
The spread, Dc 5 cmax–cmin, and the critical frequency, fc,
determine the amount of dispersion in the data. There is
no dispersion when (f/fc)4  1. The time signal, s(t), is
recovered from S(f) by the inverse FFT.
The wavelet, W(f), has a trapezoidal amplitude spectrum and a 90 phase shift. The corner frequencies of the
spectrum are 5, 10, 120, and 195 kHz, respectively, where
the signals within the 10–120 kHz band are not attenuated. The minimum and maximum phase velocities are
1000 and 2200 m/s, respectively. The effect of fc on the
dispersion and simulated time signals is illustrated in
Figure 1. The 5-kHz-fc gives rise to a sharp drop in phase
velocity within 0–50 kHz and the corresponding time
signal is simple with one cycle. The 120-kHz-fc , which
yields larger variation in phase velocity within the same
frequency band than the 200-kHz-fc , generates a more
complicated dispersive wave train. Because we want to
investigate how well the RT images dispersive energies,
we choose 120 kHz as the critical frequency.
From k to c and p to c
The Fourier f–k spectrum is transformed into the f–c
space using the relation c 5 2pf/k. Similarly, the Radon
f–p panel is mapped to the f–c space via p 5 1=c. Because
the wavenumber or slowness axis is evenly spaced, linear

Fig. 1. Simulated dispersion. (a) The dispersion curves for

three fc values: 5, 120, and 200 kHz. (b) The corresponding
trapezoidal wavelets.

interpolation is usually used to map the points from one
domain to another appropriately.
In-vitro experiment
The bone sample was a 23-cm long diaphysis of a
cervine tibia acquired from a local butcher shop. The
overlying soft tissue and the marrow of the sample were
removed and the sample was then scanned by computed
tomography (CT) to measure cortex thickness. Based
on the x-ray CT image (Fig. 2a), the top cortex had an
average thickness of 4.0 mm (minimum 5 3.6 mm,
maximum 5 4.5 mm) for the section where the transducers were deployed. The surface of the sample was
reasonably flat. The experiment setup indicates that the
bone sample was firmly held at both ends by the grabbers
of a custom-built device (Fig. 2b). Two 1-MHz angle
beam compressional wave transducers (C548, Panametrics, Waltham, MA, USA) were attached to two angle
wedges (ABWM-7T-30 deg, Panametrics). The transducer–wedge systems were positioned linearly on the
same side of the bone sample. One system acted as a
transmitter and the other as a receiver. The experiment
was carried out at 20 C (room temperature). Ultrasound
gel was applied on all contacts as a coupling agent. Constant pressure was applied to the wedges with two steel
bars to ensure good contact between interfaces. The transmitter was pulsed by a Panametrics 5800 P/R and the
recorded signals were digitized and displayed by a 200MHz digital storage oscilloscope (LeCroy 422 WaveSurfer, Chestnut Ridge, NY, USA). The digitized waveforms


Imaging in long bones using linear Radon transform d T. N. H. T. TRAN et al.

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Fig. 2. Experimental setup. (a) A sagittal computed tomography image of the cervine bone sample. Also shown is the
schematic of the transducer layout on the bone surface. The receiving transducer is moved away axially and collinearly
from the transmitter in 1-mm increment. (b) Physical setup of the experiment. Pictured is a device with grabbers at both
ends to hold the bone sample firmly in place by screws. The two steel bars are used to provide constant pressure to the
transducer/wedge systems against the bone surface.

were averaged 64 times to increase the signal-to-noise ratio. The receiver was moved away from the transmitter by
1 mm with a minimum offset of 39 mm, and 90 records
were acquired. The sampling interval, after decimation,
was 0.1 ms. The total duration of each record was
150 ms. The recorded signals formed a 1500 3 90
time–distance (t–x) matrix of amplitudes.
RESULTS
We simulated 64 time series (or records) of dispersive
wave trains to validate the performance of the RT in imaging the dispersion curve. The series are spaced 1 mm
apart and have 101 points, each with a 2-ms sampling interval. We plotted every 4 records for a total of 16 records
in Figure 3a. The records show dispersive signals of mixed
frequencies and the low-frequency components traveled
faster than the high-frequency components, which is

consistent with the simulated dispersive curve (the 120kHz-fc curve in Fig. 1a). Different frequencies have
different traveling speeds and, thus, different traveling
times. When the offset was small, the frequencies traveled
close together. As the offset increased, the difference in
traveling times became larger, and the frequencies separated, exhibiting a fanning wave train with offset. The
corresponding dispersion panels (Fig. 3b–e) show the
dependence of phase velocity (PV) resolution on the transform techniques used. Among the four, the Fourier panel
(Fig. 3b) has the worst PV resolution as the dispersive energy spreads far away from the true dispersion curve (indicated by the white dashed curve in Fig. 3) for frequencies
within 10–120 kHz. The smearing is most severe for frequencies lower than 50 kHz. The main PV spectra have

long tails and do not seem to have local extrema. The
adjoint Radon panel (Fig. 3c) has slightly better resolution
than the Fourier panel. The LSRT (Fig. 3d) improves


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Fig. 3. Simulated dispersive signals and the corresponding (f–c) dispersion panels: (a) noise-free signals; (b) 2-D fast
Fourier transform panel; (c) adjoint Radon panel; (d) damped least-squares Radon panel; (e) high-resolution Radon panel.
The true dispersion is described by the white dashed curve.

focusing better than the ART. The HRRT (Fig. 3e) focuses
the dispersive energy even better, providing a sharper image of the dispersion and superior resolution than the other
three methods. The HRRT confines the energy to a narrower band, not far from the predicted dispersion curve.
The Radon panels have alternating dark and light blue
areas, indicating side lobes with local extrema in the PV
spectra.
The transform methods imaged the PV spectrum as
broad spectra rather than narrow lines. The amount of energy spreading across a range of phase velocity values is
different for each transform method. The spreading characteristic is denoted by the PV resolution of the transform
method and can be quantified by the full-width at halfmaximum (FWHM) of the PV spectrum. The FWHM is
the full width of the PV spectrum measured at one-half
of the maximum height of the peak. Poor energy resolution or a large FWHM value means that the transform
is not capable of localizing or focusing the energy. As
an example, Figure 4 illustrates the self-normalized PV
energy spectra at 40 kHz. The FWHM values for the

FT, ART, LSRT, and HRRT are 1940, 1360, 1080, and
515 m/s, respectively. Among all, the FT has the poorest
resolution. The FWHMFT is 40% larger than the
FWHMART, 80% larger than the FWHMLSRT and 280%
larger than the FWHMHRRT. This indicates that the
ART, LSRT, and HRRT offer 40%, 80%, and 280% better
resolution, respectively, than the FT. Among the Radon
solutions, the HRRT yields 164% and 110% better resolution than the ART and LSRT, respectively.
We also applied the methods to image dispersive energies in the presence of random noise (Fig. 5). The noisy
data was generated by adding white Gaussian noise to the
noise-free signals with signal-to-noise ratio (SNR) of 10

dB. The dispersive signals were disrupted by the presence
of noise (Fig. 5a). The tracks of the imaged dispersion
were less continuous (Fig. 5b–e), but visible. As in the
noise-free case, the FT (Fig. 5b) dispersed the energy
and had difficulty confining it, thus rendering poor image
resolution, whereas the HRRT (Fig. 5d) imaged the
dispersion with enhanced resolution compared with the
other methods.
Similar to the FT, the RT also depends on the aperture. We explore here the performance of the RT in imaging data with a limited aperture (Fig. 6). Because the
HRRT has the best imaging resolution among the three
other Radon methods, we used the HRRT hereafter. We
examined PV dispersion within the frequency range
10–120 kHz where the frequency components were not
attenuated. The aperture is defined by the difference
between maximum and minimum offsets. The original

Fig. 4. Phase velocity spectra at 0.04 MHz and the corresponding full-width at half-maximum measurements.

Imaging in long bones using linear Radon transform d T. N. H. T. TRAN et al.

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Fig. 5. Simulated dispersive signals with random noise and the corresponding (f–c) dispersion panels: (a) noisy signals
with 10-dB signal-to-noise ratio; (b) 2-D fast Fourier transform panel; (c) adjoint Radon panel; (d) damped least-squares
Radon panel; (e) high-resolution Radon panel. The true dispersion is described by the white dashed curve.

reference data had 64 2-mm-spaced records with a 126mm aperture; Figure 6a illustrates the reference RP.
Next we purposely removed 6 records from the original

data to make the spatial sampling non-uniform, but kept
the aperture fixed at 126 mm. The RP (Fig. 6b) closely
resembles the original panel (Fig. 6a) without a visual

Fig. 6. Imaging simulated dispersive energy with different data apertures by the HRRT: (a) same data set as in Figure 3a,
126-mm aperture with 64 2-mm-spaced records; (b) 126-mm aperture, same data as in (a) with 6 missing records at 40, 70,
72, 100, 102, and 104 mm; (c) 124-mm aperture with the first 32 4-mm-spaced records; (d) 62-mm aperture with the first
32 2-mm-spaced records; (e) 60-mm aperture with the first 16 4-mm-spaced records; (f) 30-mm aperture with the first 16
2-mm-spaced records.


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difference. We skipped every 2 records in the original
data to incur larger spacing (Dx 5 4 mm) while keeping
the aperture at 124 mm, close to the original aperture. The

dispersion profile (Fig. 6c) looks similar to the original
profile (Fig. 6a) with a slight increase in PV spread.
Next, we considered halving the aperture to 62 mm by
taking the first 32 records of the original data. At a small
aperture, the resultant dispersion profile (Fig. 6d) suffers
energy spreading, and the smearing increases with
decreasing frequency. The same smearing effect is
observed in Figure 6e where we skipped every 4 records
of the original data to keep 16 records with a 4-mm
spacing and an aperture of 60 mm similar to the previous
case (Fig. 6d). These two data sets (Fig. 6d–e) have
similar apertures (60 mm vs. 62 mm), and their dispersion
profiles look similar even though they have different
number of records (16 vs. 32) and spacing (2 mm vs. 4
mm). Last, we lowered the aperture further to 30 mm
(half of the previous two cases) by keeping the first 16
records of the original data. The dispersion panel
(Fig. 6f) exhibits a lack of energy confinement and severe
spreading far from the true solution. Also, the imaged
dispersion track is segmented, discontinuous, and stepwise, yielding an aliased image, which might erroneously
implicate the existence of several modes. Clearly,
changes in aperture size cause more severe smearing
than reducing the number of records for a fixed aperture
size.
The cervine tibia data illustrated in Figure 7a consists of 90 records with a 89-mm aperture and 39-mm
minimum offset. The processing steps involved bandpass
filtering, linear gain, and self-normalization. The corner

Volume 40, Number 11, 2014

frequencies of the bandpass window were 0.005, 0.03,
0.8, and 1.0 MHz, while the last two processing steps
made the small late-arriving and/or far-offset signals
visible. The t–x panel exhibits mainly two types of arrivals with distinct move-outs. The first type is usually
the high-frequency high-velocity (HFHV) bulk waves
(Le et al. 2010), and the second type is the lowfrequency low-velocity (LFLV) arrivals, which are usually surface or Lamb-type guided waves (Ta et al.
2009). At close offset, the HFHV bulk waves dominated.
Between 40 and 55 mm, there was a lack of LV guided
wave energy buildup because of the short offset. The
LV signals started to become more visible after 60-mm
offset. At offsets . 100 mm, the low-velocity arrivals
took over and became quite dominant. The HV bulk
waves decayed very quickly with offset and lost their
strength after 80 mm. These observations are also evident
in the corresponding power spectral map (Fig. 7b). Between 40 and 70 mm, the data were rich in highfrequency (average 0.8 MHz) bulk waves. The data lost
the high frequencies quickly because of amplitude decay
with distance and preferential filtering caused by absorption. Between 70 and 100 mm, the frequency content of
the signals dropped to a midrange of approximate
0.35 MHz and the signals were a mixture of HV and LV
waves. After 100 mm, the 0.1-MHz signals took over
and the guided wave energies built up strongly, providing
clear evidence of the presence of late-arriving LFLV
wave modes.
Using the real data, we examined the performance of
the FT and HRRT in extracting dispersive energy when
the aperture decreased from 89 to 31 mm. There are at

Fig. 7. Cervine tibia bone sample: (a) self-normalized and linearly gained t–x signals; (b) the corresponding power
spectral density map.

Imaging in long bones using linear Radon transform d T. N. H. T. TRAN et al.

least six strong energy loci in both panels (Fig. 8a–f). To
interpret the guided modes, we simulated dispersion
curves with the commercial software package DISPERSE
Version 2.0.16i (Imperial College, London) developed by
Pavlakovic and Lowe (2001). The model was a waterfilled cylinder with a 4.4-mm-thick cortex and a 6.35mm inner radius. The density, longitudinal wave velocity,
and shear wave velocity of the cortex were 1930 kg/m3,
4000 m/s, and 2000 m/s, respectively (Le et al. 2010),
whereas the density and longitudinal wave velocity of
water were 1000 kg/m3 and 1500 m/s. Six guided modes
were identified with confidence: Fð1; 1Þ, Fð1; 5Þ, Fð1; 8Þ,
Fð1; 16Þ, Lð0; 6Þ, and Lð0; 7Þ. With the exception of the
Fð1; 1Þ mode, all modes are clearly seen in all panels.
Fð1; 1Þ was quite weak and its presence faded when
only 32 records were used (Fig. 8e–f). When the data
aperture decreased from 89 to 31 mm, the resolution of
the Fourier panels (Fig. 8a, c, e) deteriorated with significant energy smearing. For example, at 0.8 MHz, the
FWHM of Fð1; 16Þ increases from 927 m/s at an 89mm aperture (Fig. 8a) to 1935 m/s at a 31-mm aperture
(Fig. 8e), which is a greater-than twofold increase in
smearing or loss in resolution. At a 31-mm aperture (32
records), the FT lost resolution as Fð1; 8Þ, Lð0; 6Þ and
Lð0; 7Þ tended to cluster together (Fig. 8e). In contrast,
the Radon panels fared much better than the Fourier
panels. All the Radon panels exhibit good confinement
of the modal energies. When the aperture decreased, en-

2723

ergy smearing occurred but was not as severe as in the
FT case. Similarly, the FWHM also exhibited a twofold
increase from 286 to 573 mm when the aperture decreased from 89 mm (Fig. 8b) to 31 mm (Fig. 8f). Even
though only 32 records were used (Fig. 8f), the three concerned modes, Fð1; 8Þ, Lð0; 6Þ and Lð0; 7Þ, were well
separated in the Radon panels.
DISCUSSION
This study was conducted to determine the ability of
the linear Radon (or t–p) transform to image dispersive
guided wave energies in long bones, which makes our
work novel. The transform was implemented using a
least-squares strategy with Cauchy-norm regularization
that serves to improve the focusing power, that is, to
enhance resolution in the transformed domain. The proposed HRRT has also been compared with the conventional temporal–spatial Fourier transform to validate the
superiority of the method. Multichannel dispersive energy analysis requires reliable mapping of the ultrasound
data from the t–x domain to the f–k domain. The mapping
is usually performed by the conventional 2-D FFT. However, the extracted dispersion curves lack the resolution in
the transformed plane to discriminate wave modes
(Moilanen 2008; Sasso et al. 2009).
The resolving power associated with the FT is linked
to the spatial aperture of the recorded data (Moilanen

Fig. 8. Dispersion f–c panels: (a, c, e) Conventional Fourier panels; (b, d, f) Radon panels. From left to right, the numbers
of ultrasonic records are 90, 64 and 32, corresponding to 89-, 63- and 31-mm apertures, respectively. The theoretical
dispersion curves are shown in white.


2724

Ultrasound in Medicine and Biology

2008; Ta et al. 2006a). Our acquisition aperture is finite,
leading to a windowing or truncation on the x-axis.
Truncating the x-axis is equivalent to convolving the xspace with a sinc function. Consider a boxcar function, f
(x) of width a, where f (x) 5 1 for –a/2 # x # a/2 and
0 elsewhere. The width of the box, a, is the ‘‘aperture.’’
The Fourier transform of a boxcar is a sinc function, F
(k) 5 asin(ka/2)/(ka/2), with the main spectrum
bounded by the zeros: –2p/a and 2p/a. The distance
between the zeros, or zero distance, is 4p/a. As the
aperture (a) increases, the zero distance decreases, and
the width of the spectrum becomes smaller or narrower,
thus improving resolution in the k-space. This simple
illustration indicates the resolution dependence of the 2D FFT method on the spatial aperture of the acquired data.
In clinical studies of human long bones where
spatial acquisition range is restricted because of the
limited dimension of the ultrasound probe, the number
of channels, the irregularity of the acquisition surface,
and the accessibility to the skeletal site, the 2-D FFT
method may not provide sufficient resolution. The RT,
which also depends on the spatial aperture of the data,
has a smaller aperture threshold. Given the same spatial
aperture, we have found that the HRRT dispersion maps
are much better resolved than those of the conventional
2-D FFT. Although the RT has a smaller aperture tolerance than the FT method, a small 31-mm aperture
in the simulation case exhibits dispersion artifacts
(Fig. 6f), which are absent in the real data case for the
similar aperture. Nevertheless, the HRRT provides an
alternative new approach to imaging of limited-aperture
data and estimation of spectral information. The
resolving power of the HRRT will be beneficial for guided

mode identification and separation in in-vivo studies,
where the overlying soft tissue layer increases the number
of guided modes and mode density (Tran et al. 2013a).
High-resolution spectral analysis via the Burg
maximum entropy method (Marple 1987), multiple
signal classification (MUSIC) method (Schmidt 1986),
or minimum variance method (MVM) (Capon 1969)
can also be used to estimate high-resolution spectra by
applying those methods to spatial data for each temporal
frequency. The f–k energy computed by these methods
could be mapped to the f–p plane to obtain the desired energy distribution for the dispersive signals. However, the
aforementioned methods will only give a high-resolution
image of the modal energies in the f–p plane that cannot
be used to return to data (t–x) space. Our RT approach, on
the other hand, permits us to design an operator that can
be used to return to the t–x domain. This is important
because high-resolution images can be obtained in the
f–c space by plotting the absolute values of the complex
M (f, c). but can also use M (f, c) to recover D (t, x) via
the Radon forward operator, L.

Volume 40, Number 11, 2014

The acquired data contain linear (direct waves, head
waves, and surface waves) and hyperbolic (reflections)
events. By using a linear RT, we assumed all events
were linear. In consideration of the short offset configuration and a thin cortex, the close-offset portions of the
reflection events (or the t–x curves) are approximately
linear and thus, the assumption is valid. Further, a hyperbolic RT can be used if necessary (Gu and Sacchi 2009).
The HRRT maps the t–x signals to a high-resolution

dispersion diagram without requiring the spatial space to
be evenly sampled. Solving the problem using the
inverse-problem technique allows the HRRT to be used
for accurate missing data reconstruction or interpolation
in practice. To reconstruct the missing records, the offset
axis is resampled, the spatial coordinates of the missing
records are inserted and the Radon operator L is resampled to interpolate missing records or fill the data
gap. It is important to note that it is quite simple to use
the HRRT in cases where the data are irregularly sampled.
This is also true for the Fourier methods in which one
could replace the FFT with a non-uniform discrete Fourier transform (Sacchi and Ulrych 1996). However, a nonuniform discrete Fourier transform is a non-orthogonal
transform and therefore, an inversion process similar to
that outlined for the HRRT is required to have a transform
that allows us to go from t–x to f–k and return to the t–x
domain. This problem was addressed by Sacchi and
Ulrych (1996).
The HRRT is also robust in enhancing signal coherency and canceling noise. Because the amplitudes are
summed along a linear move-out, random noise is significantly attenuated because of its incoherency and randomness, but the coherent energy is reinforced, thus greatly
enhancing the SNR. Generally, solving inverse problems
takes considerable computation time because of iteration.
For the data sets used in this study, four iterations were
found to be sufficient to yield reasonable results. For
instance, it took less than 1 min to provide a dispersion
diagram in this study using a quad-core Windows 7 64bit computer with Intel Core Q6600 2.40-GHz CPU and
4-Gb RAM. Increasing the number of iterations consumes
more computation time.
The hyper-parameter, m, of the cost functions, given
by eqns (7) and (9), controls the degree of fitting the
predicted observations to the acquired data. A small mvalue leads to a solution with minimized prediction
error, but the focusing power of the transform is less

ideal. Conversely, if the m value is large, the Radon
energies will be imaged with higher resolution as the
regularization term is now emphasized, but the data
misfit will be large as well. A preferred method of
choosing the m-value is use of the L-curve (Engl and
Grever 1994), which is illustrated in Figure 9. The L-curve
is a plot of the regularization term versus the data misfit.


Imaging in long bones using linear Radon transform d T. N. H. T. TRAN et al.

Dp ,

1
rmax fmax

2725

(16)

where rmax is the offset range. The sampling intervals and
other parameters that are relevant to the simulation and
bone data sets are tabularized in Table 1.
CONCLUSIONS

Fig. 9. Example of an L-curve for the noise-free simulated data
set (Fig. 3a). The regularization and misfit terms of the L-curve
are given by eqn (9).

The optimal value of m corresponds to the ‘‘elbow’’ point

of the L-curve, where the curvature is maximal. For both
the LSRT and HRRT, we used m-values of 1000 for the
simulated data and 15,000 for the bone data.
Aliasing is associated with insufficient sampling resulting in data artifacts. To avoid aliasing, the RT should
obey the following sampling guidelines. Temporal sampling and spatial sampling are related by the Nyquist
criteria (Turner 1990). The sampling along the time
axis is governed by
1
Dt #
2fmax

(14)

where Dt is the time step and fmax is the maximum frequency present in the signals. The spatial sampling or
receiver spacing, Dx, satisfies
Dx ,

1
Pfmax

(15)

where P is the slowness range. If the spatial sampling is
not regular, Dx takes the largest spatial interval in the
data. The slowness resolution, Dp, is selected such that
Table 1. Values of the relevant parameters pertaining to
the data used in this study.
Data set

Dt

(ms)

Dp
(ms/mm)

Dx
(mm)

fmax
(MHz)

rmax
(mm)

P
(ms/mm)

Simulation
Bone sample

2
0.1

0.004
0.002

2
1

0.195

1

126
89

0.8
1

The HRRT technique provides a powerful highresolution tool to image multichannel ultrasonic dispersive energy in long bones. Applications to numerical
and in-vitro experimental data sets have demonstrated
the feasibility and robustness of the method. Although
the guided modes are distinguishable in both Fourier
and Radon panels using the in-vitro data in this study,
the HRRT illustrates a more powerful resolving power,
constraining the dispersive energies of the guided modes
within their well-delineated tracks. Therefore, the application of HRRT will be beneficial for more complex cases
where the modes come close together. The HRRT handles
smaller apertures and requires fewer records, which do
not have to be evenly spaced. In addition, the HRRT
has the added advantage of enhancing SNR by reducing
random noise. This method should be considered the
preferred method for carrying out the multichannel
dispersion analysis of ultrasonic guided wave data in
long bones, where the recording aperture is limited
because of practical constraints. The success of this study
provides a bright road map to a wide range of RT applications in the field of processing ultrasonic guided wave
data in long bones.
Acknowledgments—Tho N. H. T. Tran sincerely acknowledges the
Department of Radiology and Diagnostic Imaging, University of
Alberta, for the support of a PhD fellowship.

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APPENDIX: ESTIMATING THE
HIGH-RESOLUTION RADON SOLUTION, MHR ,
USING THE ITERATIVELY RE-WEIGHTED
LEAST-SQUARES METHOD
We want to compute the model, M, via eqn (10), which is
21

M 5 ðLH L1mQðMÞÞ LH D

(A1)

where Q is also a function of M,
QðMÞ 5

1
:
ð11M2 =s2 Þ

(A2)

Using the IRLS algorithm (Scales et al. 1988), the model at the jth
iteration, Mj , can be estimated using the previous iteration of Q, that
is, Qj – 1:
À
Á21
(A3)
Mj 5 LH L1mQj21 LH D:


Imaging in long bones using linear Radon transform d T. N. H. T. TRAN et al.
The HRRT algorithm is as follows:

The initial value Q0 is a diagonal weighting matrix calculated as
Q0;kk 5 À

1
 Á
11ðM0;k Þ2 s2

(A4)

where the damped least-squares solution provides the initial estimate
of M0 :
21

M0 5 ðLH L1mIÞ LH D:

(A5)

1:
2:
3:
4:
5:
6:
7:

procedure RADON (d, x, p, m, s, n)
Dðf ; xÞ 5 fft ðdðt; xÞÞ
for f 5 fmin, ., fmax do
L 5 exp (–i2pfxTp)
M0 ðf ; :Þ 5 ðLH L1mIÞ21 LH Dðf ; :Þ
for j 5 1, ., n do
Qj21;kk 5 ð11ðM 1ðf ;kÞÞ2 =s2 Þ
j21

For example,
21

M1 5 ðLH L1mQ0 Þ LH D:

(A6)

The iteration stops at a preset number or the convergence is reached at a
preset tolerance limit.

8:
9:
10:
11:
12:

Mj ðf ; :Þ 5 ðLH L1mQj21 Þ21 LH Dðf ; :Þ
end for (j)
end for (f)
Return MHR 5 Mn
end procedure

2727