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IMAGE COMPRESSION USING ZEROTREES OF WAVELET COEFFICIENTS

Transition from single dimension analysis to two-dimension analysis

Basics

1. A vector containing samples of an analog signal is sampled at a rate satisfying Nyquist’s rate.

2. The samples are passed through low and high half bandpass filters.

3. These filters use low pass decomposition filter coefficients and high pass decomposition filter coefficients obtained from Daubechies 4 wavelet.

4. The outputs of these filters when downsampled by a factor of 2 give approximation wavelet coefficients (LPF) and detail wavelet coefficients (HPF).

5. These approximations take the place of the original vector and are then again subjected to steps 2 through 4 to obtain as many levels of decomposition as necessary.

6. The factor to be understood is that the Nyquist’s rate is still satisfied at all levels of decomposition.

7. This leads to approximations having poor time resolution and better frequency resolution and details having better time resolution and poor frequency resolution. 

8. The result of the seventh step plays the most important role in image compression.

TRANSITION 

Time and frequency multiresolution is well understood with respect to single dimension signals. But the requirement now is to find a similarity between these two parameters and the two parameters in images, which are to be processed in order to achieve compression.

These two image parameters are space corresponding to time and color variation corresponding to frequency variation.

Let us consider an image for the analysis. Let us assume that the size of the image is 1 x 400 pixels so that in effect we can take it as a single dimensional vector.

We can see that gray color appears over a large space in the image and hence is equivalent to low frequencies occurring at almost all times over a single dimensional signal.

Colors appearing over large spaces are equated to low frequencies. These give rise to approximation coefficients.  

Black appears over small spaces in the image and is equivalent to high frequencies appearing over small intervals.  

Colors appearing over small areas and in effecting a sudden transition from the surroundings colors are equated to high frequencies. These give rise to detail coefficients.

Summary

In all practical images details are very few and localized narrowly in space. We need to know where exactly these details occur to maintain image clarity. We need not know exactly what colors because of the inherent poor resolution of the eye.

Approximations occupy a large part of practical images and we need to know to what colors do these exactly pertain to so that color gradations in the compressed image can be maintained. This is because the human eye can make out color gradation over large spaces. But here we need not exactly know where in space these colors appear because they occupy a large space and the human eye cannot make out small differences in the positions of these colors.

Wavelets give a multiresolution approach which can make use of the above factors to achieve image compression. 

Down sampling at every level of decomposition ensures that the number of wavelet coefficients never exceed the space limits of the total image.

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