Hello, everyone in this introductory session. I will cover some basic wavelet concepts
I will be primarily using a 1D example, but the same concepts can be applied to images as well
First let us review what a wavelet is
Real World Data or signals frequently exhibit slowly changing trends or oscillations punctuated with transients
On the other hand images have smooth regions interrupted by edges or abrupt changes in contrast
These abrupt changes are often the most interesting parts of the data
Both perceptually and in terms of the information they provide
The Fourier Transform is a powerful tool for data analysis
However, it does not represent abrupt changes efficiently
The reason for that is that the fourier transform represents data as a sum of sine waves which are not localized in time or space
These sine waves oscillate forever
Therefore, to accurately analyze signals and images that have abrupt changes
We need to use a new class of functions that are well localized in time and frequency
this brings us to the topic of wavelets
A wavelet is a rapidly decaying wave like oscillation that has zero mean
Unlike Sinusoids which extend to infinity a wavelet exists for a finite duration
wavelets come in different sizes and shapes
Here are some of the well-known ones
The availability of a wide range of wavelets is a key strength of wavelet analysis
To choose the right wavelet you will need to consider the application you will use it for
We will discuss this in more detail in a subsequent session
For now let us focus on two important wavelet transform concepts
Scaling and Shifting
Let us start with scaling
Say you have a signal si of t
Scaling refers to the process of stretching or shrinking the signal in time which can be expressed using this equation
S is a scaling factor which is a positive value and corresponds to how much a signal is scaled in time
The scale Factor is inversely proportional to frequency
for example
Scaling the sine wave by two results in reducing its original frequency by half or by an octave
For a wavelet there is a reciprocal relationship between the scale and the frequency with a constant of proportionality
This constant of proportionality is called the center frequency of the wavelet
This is because unlike the sine wave the wavelet has a band pass characteristic in the frequency domain
Mathematically the equivalent frequency is defined using this equation
where [Cf] is the Center frequency [of] the Wavelet s is the wavelet scale and
Delta t is the Sampling interval
Therefore when you scale a wavelet by a factor of 2 it results in reducing the equivalent frequency by an octave
Here is how a [sim] for wavelet with Center frequency of 0.07 Hertz corresponds to a sine wave of same frequency
A larger scale factor results in a stretched wavelet which corresponds to a lower frequency
A smaller scale factor results in a shrunken wavelet which corresponds to a high frequency
A stretched wavelet helps in capturing the slowly varying changes in a signal
While a compressed wavelet helps in capturing the abrupt changes
You can construct different scales that inversely correspond to the equivalent frequencies as mentioned earlier
Next we will discuss shifting
Shifting a wavelet simply means delaying or advancing the onset of the wavelet along the length of the signal
A shifted wavelet represented using this notation means that the wavelet is shifted and centered at K
We need to shift the wavelet to align with the feature we are looking for in a signal
The two major Transforms in wavelet analysis are continuous and discrete wavelet transforms
These transforms differ based on how the wavelets are scaled and shifted
More on this in the next session
But for now, you have got the basic concepts behind wavelets