Wavelets Transform Classifications

It will talk about different classifications of wavelets transform to better understand wavelet transform before applying it into real-world projects

In the last post, some basic concepts related with Wavelet transform (WT) have been discussed. In this post, we will briefly talk about the classifications of WT.

1. Classes of Wavelet Transforms

In general, WT can be divided into two broad classes:

  • continuous wavelet transforms (CWT)
  • discrete wavelet transforms (DWT)

(1) Continuous Wavelet Transforms (CWT)

CWT is composed of continuous scale parameter(????) and shift (position) parameter (????), and it can be expressed by the following expression.

Base wavelet or called other wavelet is expressed by::

By continuously varying the values of the scale parameter, ????, and the position parameter, ????, we obtain the cwt coefficients ????(????,????).

(2) Discrete Wavelet Transforms (DWT)

DWT is any wavelet transforms for which the wavelets are discretely sampled.

  • the sampling of the frequency axis corresponds to dyadic sampling
  • the scale parameter is discretized to integer powers of 2
  • The base wavelet:

As the signal ????(????) occurs at discrete integer time steps ????, the dyadic discrete wavelet transforms can be written as:

DWT decomposes a signal into a set of mutually orthogonal wavelet basis functions.

2. Other Classification Methods

Based on data Dimensions, WT can be classified into

  • 1D WT
  • 2D WT
  • nD WT

Based on decomposition Levels, WT can be classified into

  • single level/one stage WT
  • multilevel WT

Normally, Discrete Wavelet Transforms (DWT) can be divided into

  • Discrete Wavelet Transform (DWT)
  • Dual Tree Complex Wavelet Transform (DT-CWT)
  • Stationary Wavelet Transform (SWT)
  • Multiresolutiom Analysis (MRA)
  • Wavelet Packet Transform (WPT)
  • Maximum Overlap Discrete Wavelet Transform (MODWT)
  • Multiresolutiom Analysis based on MODWT (MODWTMRA)

3. Online courses

If you are interested in learning how to apply wavelet transform to real-world cases with Python step by step, you are welcome to enroll my courses:

(1) Very basic one

Practical Python Wavelet Transforms (I): Fundamentals

(2) For Real-world projects

Practical Python Wavelet Transforms (II): 1D DWT

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