  A new thresholding strategy for the estimation of a deterministic image
immersed in noise is introduced. The threshold is combined with a wavelet
decomposition, where the wavelet coefficient of the image at any fixed value of
the decomposition index is estimated, via thresholding the observed coefficient
depending on the value of both the magnitude of the observed coefficient as
well as the magnitudes of coefficients of a set of additional images calculated
from the observed image. The additional set of images is chosen so that the
wavelet transforms of the full set of images have suitable deterministic and
joint stochastic properties at a fixed scale and position index. Two different
sets of additional images are suggested. The behaviour of the threshold
criterion for a purely noisy image is investigated and a universal threshold is
determined. The properties of the threshold for some typical deterministic
signal structures are also given. The risk of an individual coefficient is
determined, and calculated explicitly when the universal threshold is used, and
some typical deterministic signal structures. The method is implemented on
several examples and the theoretical risk reductions substantiated.
