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. 2010 Nov;57(11):2437-49.
doi: 10.1109/TUFFC.2010.1710.

Complex Principal Components for Robust Motion Estimation

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Free PMC article

Complex Principal Components for Robust Motion Estimation

F William Mauldin Jr et al. IEEE Trans Ultrason Ferroelectr Freq Control. .
Free PMC article

Abstract

Bias and variance errors in motion estimation result from electronic noise, decorrelation, aliasing, and inherent algorithm limitations. Unlike most error sources, decorrelation is coherent over time and has the same power spectrum as the signal. Thus, reducing decorrelation is impossible through frequency domain filtering or simple averaging and must be achieved through other methods. In this paper, we present a novel motion estimator, termed the principal component displacement estimator (PCDE), which takes advantage of the signal separation capabilities of principal component analysis (PCA) to reject decorrelation and noise. Furthermore, PCDE only requires the computation of a single principal component, enabling computational speed that is on the same order of magnitude or faster than the commonly used Loupas algorithm. Unlike prior PCA strategies, PCDE uses complex data to generate motion estimates using only a single principal component. The use of complex echo data is critical because it allows for separation of signal components based on motion, which is revealed through phase changes of the complex principal components. PCDE operates on the assumption that the signal component of interest is also the most energetic component in an ensemble of echo data. This assumption holds in most clinical ultrasound environments. However, in environments where electronic noise SNR is less than 0 dB or in blood flow data for which the wall signal dominates the signal from blood flow, the calculation of more than one PC is required to obtain the signal of interest. We simulated synthetic ultrasound data to assess the performance of PCDE over a wide range of imaging conditions and in the presence of decorrelation and additive noise. Under typical ultrasonic elasticity imaging conditions (0.98 signal correlation, 25 dB SNR, 1 sample shift), PCDE decreased estimation bias by more than 10% and standard deviation by more than 30% compared with the Loupas method and normalized cross-correlation with cosine fitting (NC CF). More modest gains were observed relative to spline-based time delay estimation (sTDE). PCDE was also tested on experimental elastography data. Compressions of approximately 1.5% were applied to a CIRS elastography phantom with embedded 10.4-mm-diameter lesions that had moduli contrasts of -9.2, -5.9, and 12.0 dB. The standard deviation of displacement estimates was reduced by at least 67% in homogeneous regions at 35 to 40 mm in depth with respect to estimates produced by Loupas, NC CF, and sTDE. Greater improvements in CNR and displacement standard deviation were observed at larger depths where speckle decorrelation and other noise sources were more significant.

Figures

Fig. 1
Fig. 1
Three source signals with variable delay characteristics (panel A) and weightings (panel B) are summed to form a simulated ensemble of echo data, X, (panel C) which models decorrelation arising from differential motion across the point spread function (psf). Resulting echo data, X, is windowed, kwX as denoted by the rectangle. PCDE operates on this subset of echo data to compute principal components (PCs), which correspond to different source signals. In panel D, the PCDE estimated displacements of the sources signals, Δτ̂, are illustrated. In this paper, we are only concerned with the most energetic source signal and so only the first PC was computed.
Fig. 2
Fig. 2
Regions of interest used to compute contrast to noise ratio (CNR) from elastograms. The target region, t was used along with a background region that was centered at either 20 mm in depth, b1, or centered at 40 mm in depth, b2. Regions of interest were approximately 5 mm in diameter.
Fig. 3
Fig. 3
Average bias (left) and standard deviation (right) for each estimator over 1000 trials when subsample shifts are varied and the signal model does not include electronic noise or decorrelation.
Fig. 4
Fig. 4
Row (a) illustrates average estimated displacement (left) and standard deviation (right) for each estimator over 1000 trials when only out-of-beam decorrelation is simulated and differential motion is not present (α = 0). Resulting echo correlation was varied between 0.8 and 1.0. In row (b), average estimated displacement (left) and standard deviation (right) are illustrated when both out-of-beam decorrelation and differential motion decorrelation are simulated. Differential motion weighting was set according to default parameters in Table I (α = 1). The dashed line is defined as the ‘true’ displacement, which corresponds to the peak displacement in the beam. For illustrative purposes, the x-axis is not a linear scale.
Fig. 5
Fig. 5
Row (a) illustrates average estimated displacement (left) and standard deviation (right) for each estimator over 1000 trials when only differential motion decorrelation is simulated with no out-of-beam decorrelation (ρ = 1.0). In row (b), average estimated displacement (left) and standard deviation (right) for each estimator over 1000 trials are illustrated when both out-of-beam decorrelation and differential motion decorrelation are simulated. The dashed line is defined as the ‘true’ displacement, which corresponds to the peak displacement in the beam. Echo correlation resulting from out-of-beam decorrelation was set according to default parameters in Table I (ρ = 0.98). In all cases, the differential motion weighting coefficient was varied between 0 and 2.
Fig. 6
Fig. 6
Average estimated displacement (left) and standard deviation (right) for each estimator over 1000 trials when subsample shift is varied between 0 and 1 sample. The dashed line is defined as the ‘true’ displacement, which corresponds to the peak displacement in the beam. The default simulation conditions of Table I were used such that both out-of-beam decorrelation and differential motion decorrelation were simulated.
Fig. 7
Fig. 7
Average estimated displacement (left) and standard deviation (right) for each estimator over 1000 trials when simulated echo shifts are varied between 1 and 3 samples. The dashed line is defined as the ‘true’ displacement, which corresponds to the peak displacement in the beam. The default simulation conditions of Table I were used such that both out-of-beam decorrelation and differential motion decorrelation were simulated.
Fig. 8
Fig. 8
Average estimated displacement (left) and standard deviation (right) for each estimator over 1000 trials when simulated SNR is varied between 0 and 45 dB. The dashed line is defined as the ‘true’ displacement, which corresponds to the peak displacement in the beam. The default simulation conditions of Table I were used such that both out-of-beam decorrelation and differential motion decorrelation were simulated.
Fig. 9
Fig. 9
Average estimated displacement (left) and standard deviation (right) for each estimator over 1000 trials when fractional bandwidth is varied between 5% and 100%. The dashed line is defined as the ‘true’ displacement, which corresponds to the peak displacement in the beam. The default simulation conditions of Table I were used such that both out-of-beam decorrelation and differential motion decorrelation were simulated.
Fig. 10
Fig. 10
Average estimated displacement (left) and standard deviation (right) for each estimator over 1000 trials when kernel length is varied between 0.5 and 10 periods. The dashed line is defined as the ‘true’ displacement, which corresponds to the peak displacement in the beam. The default simulation conditions of Table I were used such that both out-of-beam decorrelation and differential motion decorrelation were simulated.
Fig. 11
Fig. 11
Experimental elastography displacement maps formed using a) principal component displacement estimation, b) spline time delay estimation (sTDE), c) the Loupas algorithm, and d) normalized cross-correlation with cosine fitting (NC CF). Experiments used a CIRS Model 049A phantom with type I lesion of approximately 8.7 kPa and background stiffness of approximately 25.2 kPa.
Fig. 12
Fig. 12
Elastograms formed using principal component displacement estimation (column I), spline time delay estimation (column II), the Loupas algorithm (column III), and normalized cross-correlation with cosine fitting (column IV). Rows are elastograms of lesion type I (row aA), lesion type II (row b), and lesion type III (row c). In all instances, kernel length was 3 periods with 90% overlap.
Fig. 13
Fig. 13
Average contrast to noise ratio (CNR) values when the background region is centered at a) 20 mm and b) 40 mm. Error bars illustrate the mean CNR plus or minus one standard deviation over 3 trials.
Fig. 14
Fig. 14
Time to compute one displacement profile averaged over 1000 trials. Ensemble lengths are varied between 2 and 300 A-lines. ‘PCDE – Eig’ denotes PCDE estimates when MATLAB function ‘eig’ is used to compute eigenvectors. Similarly, ‘PCDE – Eigs’ denotes PCDE estimates when MATLAB function ‘eigs’ is used to compute eigenvectors.

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