Minimax threshold for denoising complex signals with Waveshrink

For the problem of signal extraction from noisy data, Waveshrink has proven to be a powerful tool, both from an empirical and an asymptotic point of view. Waveshrink is especially efficient at estimating spatially inhomogeneous signals. A key step of the procedure is the selection of the threshold parameter. Donoho and Johnstone (1994) propose a selection of the threshold based on a minimax principle. Their derivation is specifically for real signals and real wavelet transforms. In this paper, we propose to extend the use of Waveshrink to denoising complex signals with complex wavelet transforms. We illustrate the problem of denoising complex signals with an electronic surveillance application.