Haar小波问题

Question

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 question is:

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.

Answer 1

Haar basis is a orthogonal basis, that contains not all data about one pixel (this is a canonical basis), but data about localized areas. So original image is a supersposition of all images that obtained after you will find all dot product between original image and haar basises (if you have 8x8 picture you need 64 basis functions) so, by applying analysis formula you will find coefficitents. Linear combination of haar basis and this coefficients will give you the original image. But because of the properties of Haar Basis, when you cut half of this coefficients you do not cut half of your image, you just make that image less detalized.

as it shown here