Estimation of Time-Scaling Factor for Ultrasound Medical Images Using the Hilbert Transform
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Research Article Estimation of Time-Scaling Factor for Ultrasound Medical Images Using the Hilbert Transform ´ emie ´ Jer Fromageau,1 Herve´ Liebgott,2 Elisabeth Brusseau,2 Didier Vray,2 and Philippe Delachartre2 1 Laboratory
of Biorheology and Medical Ultrasonics (LBUM), University of Montreal Hospital, 2099 Pavilion J. A. de S`eve, Montr´eal, QC, Canada H2L 2W5 2 Centre de Recherche et d’Applications en Traitement de l’Image et du Signal (CREATIS), CNRS UMR 5515, Inserm U 630, INSA de Lyon, 7 Avenue Jean Capelle, 69 621 Villeurbanne Cedex, France Received 20 April 2006; Revised 20 September 2006; Accepted 20 September 2006 Recommended by Tan Lee A new formulation for the estimation of the time-scaling factor between two ultrasound signals is presented. The estimator is derived under the assumptions of a small time-scaling factor and signals with constant spectrum over its bandwidth. Under these conditions, we show that the proposed approach leads to a simple analytic formulation of the time-scaling factor estimator. The influences of an increase of the time-scaling factor and of signal-to-noise ratio (SNR) are studied. The mathematical developments of the expected mean and bias of the estimator are presented. An iterative version is also proposed to reduce the bias. The variance is calculated and compared to the Cramer-Rao lower bound (CRLB). The estimator characteristics are measured on flat-spectra simulated signals and experimental ultrasound scanner signals and are compared to the theoretical mean and variance. Results show that the estimator is unbiased and that variance tends towards the CRLB for SNR higher than 20 dB. This is in agreement with typical ultrasound signals used in the medical field, as shown on typical examples. Effects of the signal spectrum shape and of the bandwidth size are evaluated. Finally, the iterative version of the estimator improves the quality of the estimation for SNR between 0 and 20 dB as well as the time-scaling factor estimation validity range (up to 15%). Copyright © 2007 J´er´emie Fromageau et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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INTRODUCTION
The interest of the time-scale variation between two signals was shown in many applications. As an example, in radar or sonar systems, the Doppler shift, between the received and the transmitted signals, allows the estimation of the relative velocity of a target [1]. In medical imaging, the Doppler shift enables the estimation of blood flow velocity [2] or strain of soft biological tissues by comparing a pair of ultrasonic signals acquired before and after tissue compression [3, 4], the time-scaling factor then represents the tissue strain. In most applications, this time-scaling factor is considered small. For example, the target velocity is small compared to the wave speed. Similarly, in tissue strain estimation, the compression applied to the tiss
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