Wavelets transform overview
What is a Wavelet?
Wavelets are dilations and translations of a specifically shaped function the so-called Mother Wavelet. The mother wavelet has finite support
meaning that it is non-zero for only a short distance. In other words, the
mother wavelet function, W0 (x) or W0 (x,y), is mostly zero except for a
narrow region where it departs from zero.
Dilating the wavelet widens or increases the size of the region where the
function is non-zero, and translation simply means moving the function around
so it's centered on a different portion of the data. In terms of equations if
W0(x) is non-zero for 0 < x < 1 then Wab(x) = W0( (x-b)/a ) is non-zero for b <
x < b + a. Wab(x) is the mother wavelet translated by 'b' and dilated by 'a'. For
2D wavelets there will be two translation parameters, bx and by, but there is
still only one dilation parameter 'a' i.e. Wabxby(x,y) = W0( (x-bx)/a, (y-by)/a )
A wavelet's shape will always remain the same for a given wavelet - only it's
non-zero width and position changes.
The shape of a given wavelet is generally arbitrary though certain wavy
shapes are more commonly used. Some shapes such as those of the
Daubechies wavelets can be quite unexpected.
Where do Wavelets come from?
Most wavelets, including the Daubechies wavelets, come from a
corresponding Multiresolution Analysis. For example, average some data
on scale L, and then average that same data on scale L/2. The difference
between the two averaged signals is a wavelet transform on (dilation) scale L.
A wavelet transform is basically a band-pass signal since variations in the
data on scales greater than L and smaller than L/2 are mostly removed. How
the data is "averaged" is what determines the Multiresolution Analysis and
ultimately the form of the underlying wavelet.
For example, a (non-overlapping) boxcar average results in a piece-wise
constant multiresolution analysis and the Haar wavelet, so the Haar wavelet
transform on dilation scale L is equivalent to a (non-overlapping) boxcar
average of the data on scale L/2 subtracted from the data boxcar averaged on
scale L.
Other wavelets are designed (or invented) in order to match up with particular
"shapes" or patterns of variation in the data, and may have no particular
meaning beyond that.
How are Wavelets used?
Wavelets are best used when the data variations are intermittent or nonperiodic. For example, an early, common use of wavelets was for seismic
analysis where the data representing the movement of the earth's crust
includes highly sporadic fluctuations.
Wavelets can be used to determine where in the data and on what scale the
strongest variations in the data occur.
Wavelets are used to identify events in the data, which in turn can be
sampled and then studied in more detail.
The Multiresolution Analysis aspect of wavelets is extremely useful for
efficiently visualizing data variations on many different scales. This allows for
zooming in and out - in to see a leaf on a tree and then back out to see the
whole forest. This is helpful when there are millions of data points and
variations occur on all of the resolved scales - generally the case in nature.
Wavelets are useful for calculating averaged global spectra because they
are inherently pre-smoothed. Meaning that if you've ever calculated raw
Fourier spectra they usually come out quite noisy and require some sort of
band averaging in order to easily interpret. Wavelet spectra are inherently
"band-averaged."
One useful application of Wavelets includes Wavelet Cospectra wherein
wavelet transforms are applied simultaneously to multiple variables. Wavelet
Cospectra and more generally cross-wavelet transforms can be used to
identify localized correlations between different variables.
Wavelet applications in medical signal analysis
Over recent years, wavelet transforms have played an increasingly important
role in the medical signal analysis. Wavelet transform analysis has been
applied to a wide variety of biomedical signals including: the ECG, EEG,
EMG, Echocardiograms, MRI Images, clinical sounds-arterial bruits, heart
sounds, breath sounds, respiratory patterns, blood pressure trends, and DNA
sequences.
The two plots below show a short segment of ECG containing normal sinus
rhythm together with its associated wavelet transform scalogram. The QRS
complex of the waveform is evident from the conical structures in the
scalogram, converging to the high frequency components of the RS spike.
The P and T waves are also labelled in the plot. This figure highlights the
wavelet transform's ability to pick out short duration, high frequency
components in the time-frequency plane. An equivalent short time Fourier
transform (STFT) spectogram smears this short duration information due to its
fixed width window.
The following plots show an ECG signal before and after the wavelet
transform using our software.