Complex principal components for robust motion estimation

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-mmdiameter 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.