Image Processing -- Haar Transfrom

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The Haar Wavelet Transform divides an array of N sample points into two smaller arrays: one for averaging adjacent points and another for computing finite differences. This process can be recursively applied, resulting in a low-resolution averaged image and multiple levels of the difference array, which aids in reconstructing the original data. Despite maintaining the same number of data points post-transformation, questions arise regarding its memory-saving capabilities and its role in compression. The discussion highlights that the down-sampled image and the multiple levels of the difference array may simplify certain operations. Additionally, there is curiosity about deriving filtering formulas, as the discrete wavelet transform typically requires solving matrix equations for wavelet basis coefficients. A referenced article provides further insights into the application of Haar Transformation in image compression.
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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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