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

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Ultrasound in Med & Biol., Vol 40, No 11, pp 2715–2727, 2014 Copyright Ó 2014 World Federation for Ultrasound in Medicine & Biology Printed in the USA All rights reserved 0301-5629/$ - see front matter http://dx.doi.org/10.1016/j.ultrasmedbio.2014.05.021 d 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 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 le@ualberta.ca) 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: lawrence.le@ualberta.ca 2715 2716 Ultrasound in Medicine and Biology 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 N 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 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 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 t 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 K 21 X mðt t2pk xn ; pk Þ; n 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 K 21 X Mðf ; pk Þe2i2pfpk xn (2) k50 where f is the frequency In matrix notation, eqn (2) becomes D LM where L is the linear Radon operator 2iup x e 0 / e2iupK21 x0 L54 « « 2iup0 xN21 2iupK21 xN21 e / e (3) 2717 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 LM1N: (6) We seek a Radon solution or model, M, which minimizes the following cost or objective function in a least-squares sense: J kLM2Dk 1mkMk : 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 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 kLM2Dk 1m K 21 X À  Á ln 11Mk2 s2 (9) k50 with u 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 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 LH D M HR ðLH L1mQðMÞÞ LH D (5) 21 (10) 2718 Ultrasound in Medicine and Biology Volume 40, Number 11, 2014 where QðMÞ is a diagonal weighting matrix, that is, Qkk 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  Sðf Þ Wðf Þe 2i2pf À x 2t cðf Þ Áà (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 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 cmax–cmin, and the critical frequency, fc, determine the amount of dispersion in the data There is no dispersion when (f/fc)4  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 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 2pf/k Similarly, the Radon f–p panel is mapped to the f–c space via p 1=c Because the wavenumber or slowness axis is evenly spaced, linear Fig 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 3.6 mm, maximum 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 2719 Fig 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 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 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 mm apart and have 101 points, each with a 2-ms sampling interval We plotted every 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 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 2720 Ultrasound in Medicine and Biology Volume 40, Number 11, 2014 Fig 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 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 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 2721 Fig 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 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 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 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 2722 Ultrasound in Medicine and Biology difference We skipped every records in the original data to incur larger spacing (Dx 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 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 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 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 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) for –a/2 # x # a/2 and elsewhere The width of the box, a, is the ‘‘aperture.’’ The Fourier transform of a boxcar is a sinc function, F (k) 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 to provide a dispersion diagram in this study using a quad-core Windows 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 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 , 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 CONCLUSIONS Fig 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 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 , 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 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 0.1 0.004 0.002 0.195 126 89 0.8 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 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 REFERENCES Alleyne D, Cawley P A 2-dimensional Fourier-transform method for the measurement of propagating multimode signals J Acoust Soc Am 1991;89:1159–1168 Camus E, Talmant M, Berger G, Laugier P Analysis of the axial transmission technique for the assessment of skeletal status J Acoust Soc Am 2000;108:3058–3065 Capon J High-resolution frequency-wavenumber spectrum analysis Proc IEEE 1969;57:1408–1418 Cawley P, Lowe MJS, Alleyne DN, Pavlakovic B, Wilcox PD Practical long range guided wave testing: applications to pipes and rail Mater Eval 2003;61:66–74 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Lou E Spectral ratio method to estimate broadband ultrasound attenuation of cortical bones in vitro using multiple reflections Phys Med Biol 2007;52: 5855–5869 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 ðLH L1mQðMÞÞ LH D (A1) where Q is also a function of M, QðMÞ : 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 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 À  Á 11ðM0;k Þ2 s2 (A4) where the damped least-squares solution provides the initial estimate of M0 : 21 M0 ðLH L1mIÞ LH D: (A5) 1: 2: 3: 4: 5: 6: 7: procedure RADON (d, x, p, m, s, n) Dðf ; xÞ fft ðdðt; xÞÞ for f fmin, , fmax L exp (–i2pfxTp) M0 ðf ; :Þ ðLH L1mIÞ21 LH Dðf ; :Þ for j 1, , n Qj21;kk ð11ðM 1ðf ;kÞÞ2 =s2 Þ j21 For example, 21 M1 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 ðLH L1mQj21 Þ21 LH Dðf ; :Þ end for (j) end for (f) Return MHR Mn end procedure 2727 ... 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 2721 Fig Simulated dispersive signals... 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... 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