The 2-D Fourier transform analysis of multichannel signals is a straightforward method to extract the dispersion curves of guided modes. Basically, the time signals recorded at several positions along the waveguide are converted to the wavenumber-frequency space, so that the dispersion curves (i.e., the frequency-dependent wavenumbers) of the guided modes can be extracted by detecting peaks of energy trajectories. In order to improve the dispersion curve extraction of low-amplitude modes propagating in a cortical bone, a multiemitter and multireceiver transducer array has been developed together with an effective singular vector decomposition (SVD)-based signal processing method. However, in practice, the limited number of positions where these signals are recorded results in a much lower resolution in the wavenumber axis than in the frequency axis. This prevents a clear identification of overlapping dispersion curves. In this paper, a sparse SVD (S-SVD) method, which combines the signal-to-noise ratio improvement of the SVD-based approach with the high wavenumber resolution advantage of the sparse optimization, is presented to overcome the above-mentioned limitation. Different penalty constraints, i.e., l -norm, Frobenius norm, and revised Cauchy norm, are compared with the sparse characteristics. The regularization parameters are investigated with respect to the convergence property and wavenumber resolution. The proposed S-SVD method is investigated using synthetic wideband signals and experimental data obtained from a bone-mimicking phantom and from an ex-vivo human radius. The analysis of the results suggests that the S-SVD method has the potential to significantly enhance the wavenumber resolution and to improve the extraction of the dispersion curves.
Broadband ultrasound attenuation (BUA) is commonly measured by the spectral ratio method. Conventionally BUA is measured in transverse transmission mode where ultrasound signal is recorded with and without the sample. The spectral ratio method was extended to estimate nBUA (BUA normalized by thickness) in axial transmission mode using spectral amplitudes of the primary reflection and multiple reflection, which echoes more than once between the material interfaces within a layer. We performed three experiments. First, reflections were numerically simulated to verify the accuracy of the method. We then applied the method to estimate attenuation of silicon rubber and the cortex of a bovine femur. The center frequency of the transducers is 2.25 MHz. We obtained 93% accuracy for a simulated data set with 10% random noise after bandpass filtering. For the silicon rubber, 15 measurements were collected and the mean attenuation was 6.33 +/- 0.19 dB MHz(-1) cm(-1). For the bovine bone, eight measurements were performed in the middle portion of the femur. The mean attenuation was 4.91 +/- 0.65 dB MHz(-1) cm(-1) and compared well with those reported in the literature. The results demonstrate that the proposed method has the potential to provide a quick, reliable and robust cortical attenuation assessment in vivo.
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