Once one has the wavelet tools for the univariate case available, the generalization of these methods to the multivariate case is not very complicated. Besides specific higher-dimensional wavelet bases which we do not consider here, one can easily construct multidimensional bases by taking tensor products of one-dimensional basis functions. There exist two important schemes of constructing such bases.
First, as it is usually done, one can devise a multivariate
multiresolution analysis in the following way: At a given resolution
scale one takes tensor products of dyadically translated father and
mother wavelets from a one-dimensional multiresolution analysis.
This gives a -dimensional isotropic wavelet basis with a
one-dimensional scale index and a
-dimensional location index.
Second, one can also take all possible tensor products of
univariate basis functions. This yields a -dimensional anisotropic
wavelet basis with a
-dimensional scale index and a
-dimensional location index.
The isotropic basis provides a -dimensional multiresolution
analysis in the usual sense. On first sight it seems to be more
appealing than the anisotropic basis and it is almost exclusively
used in statistics. Appropriate wavelet estimators based on the
isotropic basis can attain minimax rates of convergence in isotropic
smoothness classes, which justifies its use in statistics.
Inbetween there exists also the possibility of a smooth transition
from an iso- to an anisotropic basis. This method is implemented in
XploRe
(see the command
fwt2
).
However, it was shown by Neumann and Sachs (1997) in the two-dimensional case that the isotropic basis is not really able to adapt to different degrees of smoothness in different directions. Expressed in terms of the kernel-estimator language, a projection estimator using basis functions from this basis cannot mimic a multivariate kernel estimator based on a product kernel with different (directional) bandwidths. It is shown by Neumann and Sachs (1997) and Neumann (1996a) that estimators based on the anisotropic wavelet basis can attain minimax rates of convergence in anisotropic smoothness classes. Moreover, thresholded estimators in this basis are able to recognize a lower-dimensional structure in the data. For example, if the underlying function is of additive or multiplicative structure in certain univariate components, then appropriately thresholded estimators attain one-dimensional rates of convergence. These results obtained under structural restrictions can be extended to really nonparametric functional classes. Smoothness classes with dominating mixed derivatives also allow nearly one-dimensional rates of convergence.
A quite natural application of this methodology (to the particular problem of estimating the time-varying spectral density of a locally stationary process) can be found in Neumann and Sachs (1997). In this case the two axes on the plane, time and frequency, have a specific meaning. Accordingly, one cannot expect the same degrees of smoothness in both directions. Hence, the use of the anisotropic basis seems to be more natural than the use of the isotropic one.
In principle, this methodology derived in quite a general setting can also be applied to image denoising. Assume we have a certain grey-scale image given on a grid of pixels. Assume further that this image has been blurred by some additive noise. Now one can use wavelet thresholding to perform some (partial) denoising. Since the anisotropic basis is never essentially worse but sometimes better than the isotropic basis, it seems to be advisable to employ the former construction.
Note, however, that neither the anisotropic nor the isotropic basis is devised for the specific purpose of image reconstruction. Denoising in one of these bases is an all-purpose method which can be applied to a wide variety of problem settings. We do not make the attempt to compare this method with the many denoising techniques developed by engineers for the specific purpose of image reconstruction. We think that some of these special-purpose methods will certainly outperform our simple all-purpose approach.
The display above shows in the top row three versions of an image. The left image is the original, the middle is the original one plus noise and the right one is the denoised image. The threshold value is chosen by an experienced data analyst. The lower row shows the magnitude of the ordered wavelet coeffcients. You can try to change the noise level and threshold of the wavelet basis with the following interactive menu.