Associate Professor, University of New South Wales

Wavelet methods are very popular in density estimation,
non-parametric regression and in signal analysis because
of their ability to adapt to the local irregularities of the curve.
When estimating spatially inhomogeneous curves, wavelet
methods significantly outperform other traditional nonparametric methods.
Typical wavelet estimators are **non-linear** threshold-type
estimators. They attain (possibly within a factor that is only logarithmic
in the sample size) asymptotically optimal rates simultaneously for
large class of error measures and function spaces.

My research on wavelet methods has been focused on improving their flexibility in curve estimation. I have been looking for a finer balance between stochastic and approximation terms in the mean integrated squared error decomposition. I have also been dealing with data-based recommendations for choosing the tuning parameters in the estimation procedure, with implementing some modifications in the wavelet estimators as to make them satisfy some additional shape constraints like non-negativity, integral equal to one etc.

I have also implemented an online adaptive procedure of multiple sampling
which allows for economic sampling of signals. The technique involves increasing
the sampling rate when high-frequency terms are incorporated in the wavelet
estimator, and decreasing it when signal complexity is judged to have decreased.
The size of the wavelet coefficients at suitable resolution levels is used in deciding
how and when to switch rates.
The figure here (pdf)
shows in its first panel the true signal. An additive Gaussian white noise with constant
variance was added at a rate of 100 Hz to it and the goal was online
recovery of the signal. In the second panel, our **dual-rate** sampling procedure
has been applied. It samples at a low rate of only 16 Hz most of the time.
However, when guided by the size of the wavelet coefficients, a highenough
frequency is indicated, the procedure switches automatically to a high sampling rate of 100 Hz.
This has happened near time points of 58 and of 75. When the high frequency
fluctuations are over, the procedure switches back automatically to low rate
sampling. The superimposed dashed line in the second panel indicates sampling
rate as a function of time. Where the line is at level 0 or 0.5 the rate was low or
high, respectively. The third panel shows the performance of the **constant-rate**
sampler that uses the **same** number of sampling points like our dual-rate sampler.
It can be seen that, for the same cost of sampling, the dual-rate sampler can resolve much
better the high-frequency aberrations near the time points of 58 and of 75. Sometimes,
such high-frequency aberrations are the most important feature in the signal.

My recent interest in wavelet methods is in applying them for image analysis.