Even though some individuals made slight advances in the field of wavelets
from the 1930’s to the 1970’s, the next major advancements came from Jean
Morlet around the year 1975. In fact, Morlet was the first researcher
to use the term “wavelet” to describe his functions. More specifically,
they were called “Wavelets of Constant Slope."
Before 1975, many researchers had pondered over the idea of Windowed Fourier
Analysis (mainly a man named Dennis Gabor). This idea allowed us
to finally consider things in terms of both time and frequency. Windowed
Fourier Analysis dealt with studying the frequencies of a signal piece
by piece (or window by window). These windows helped to make the
time variable discrete or fixed. Then different oscillating functions
of varying frequencies could be looked at in these windows.
Morlet, also a graduate of Ecole Polytechnique, tried Windowed Fourier
Analysis while working for an oil company named Elf Aquitaine. Oil
companies searched for underground oil by sending impulses into the ground
and analyzing their echoes. These echoes could be analyzed to tell
how thick a layer of oil underground would be. Fourier Analysis and
Windowed Fourier Analysis were used to analyze these echoes; however, Fourier
Analysis was a time-consuming process so Morlet began to look elsewhere
for a solution.
When he worked with Windowed Fourier Analysis he discovered that keeping
the window fixed was the wrong approach. He did exactly the opposite.
He kept the frequency of the function (number of oscillations) constant
and changed the window. He discovered that stretching the window
stretched the function and scrunching or squeezing the window compressed
the function. In fact, you can see the close resemblance between
the sine functions used in Fourier Analysis and the Morlet wavelets.
This concept
of Morlet’s was touched on briefly with the Haar wavelets from earlier.
The 0th daughter wavelet was simply a squeezed version of her mother.
If we start with the mother wavelet on the interval [0,1], we can squeeze
the window in half and obtain the 0th daughter wavelet. It has the
same shape, but on a smaller interval [0, ½].
Morlet
had made quite an impact on the history of wavelets; however, he wasn’t
satisfied with his efforts by any means. In 1981, Morlet teamed up
with a man named Alex Grossman. Morlet and Grossman worked on an
idea that Morlet discovered while experimenting on a basic calculator.
The idea was that a signal could be transformed into wavelet form and then
transformed back into the original signal without any information being
lost. When no information is lost in transferring a signal into wavelets
and then back, the process is called lossless. Morlet’s concept is
something that beginning wavelet students do all the time, but rarely think
of how big of a breakthrough it was. Morlet and Grossman’s efforts
with this concept were a complete success. The only resources they
needed were a personal computer and the brainpower of two incredible mathematicians
(even though they didn’t consider themselves mathematicians). Since
wavelets deal with both time and frequency, they thought a double integral
would be needed to transform wavelet coefficients back into the original
signal. However, in 1984, Grossman found that a single integral was
all that was needed.
While working
on this idea, they also discovered another interesting thing. Making
a small change in the wavelets only causes a small change in the original
signal. This is also used often with modern wavelets. In data
compression, wavelet coefficients are changed to zero to allow for more
compression and when the signal is recomposed the new signal is only slightly
different from the original. If slightly changing wavelet coefficients
would have introduced a big error after recomposing back to the original
signal, data compression today would be a much more difficult task.