On multivariate estimation by thresholding

Despite their simplicity, scalar threshold operators effectively remove additive white Gaussian noise from wavelet detail coefficients of many practical signals. This paper explores the use of multivariate estimators that are almost as simple as scalar threshold operators. Sendur and Selesnick have recently shown the effectiveness of joint threshold estimation of parent and child wavelet coefficients. This paper discusses analogous results in two situations. With a frame representation, a simple joint threshold estimator is derived and it is shown that its generalization is equivalent to a type of l(1)-regularized denoising. Then, for the case where multiple independent noisy observations are available, the counter-intuitive results by Chang, Yu, and Vetterli on combining averaging and thresholding are explained as a fortuitous consequence of randomization.