Image Processing -- Haar Transfrom

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SUMMARY

The Haar Transform, specifically the Haar Wavelet Transform, subdivides an array of N sample points into two arrays: one for averaging adjacent points and another for computing finite differences. This process can be recursively applied to generate a low-resolution averaged image and multiple levels of the difference array, facilitating data recovery. While the transform retains the original number of data points, it aids in image compression and simplifies certain operations on down-sampled images.

PREREQUISITES
  • Understanding of Haar Wavelet Transform principles
  • Familiarity with image processing concepts
  • Knowledge of recursive algorithms
  • Basic grasp of finite difference methods
NEXT STEPS
  • Research the mathematical foundations of Haar Wavelet Transform
  • Explore image compression techniques using discrete wavelet transforms
  • Learn about the implementation of recursive algorithms in image processing
  • Investigate filtering formulas and matrix equations for wavelet basis functions
USEFUL FOR

Image processing professionals, data scientists, and software developers interested in efficient image compression techniques and the application of wavelet transforms in data recovery.

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I'm looking at this link: HaarWaveletTransform. Given an array of N sample points, it subdivides the array into two arrays of size N/2:

Array1: Averages adjacent sample points. Array2: Computes a finite difference between sample points.

You can then apply recursively k many times. In the end you will get a low resolution averaged image and multiple levels of Array2 which help invert the operation to recover the original data.

After the transform you still have as many data points as you originally had. So my questions are:

  1. How does this save memory? I thought this was supposed to help with compression?
  2. What is the point? Are some operations easier when you have the down sampled image and multiple levels of Array2?
  3. How do you get these filtering formulas? I thought for discrete wavelet transform you would have to solve some matrix equation to compute the coefficients for the wavelet basis functions.
 
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