Wavelet transform coding

An easy way to construct a multi-dimensional (e.g. 2-D) wavelet transform is, for example, to implement the tensor products of the 1-D counterparts. That is, we apply the 1-D wavelet transform separately along one dimension at a time. This, as we shall see shortly, results in one scaling function and three different mother wavelet functions. 

The sub-bands in each successive dyadic decomposition step are generally labelled simply as Il,li Il.h. Ih.l and Ih.h ( L for low frequency, and 

Wavelet-based software packages generate the sub-bands in the form of a grouped display of the smoothed and detailed sub-bands. For example, the grouped display of the sub-bands in a single decomposition step as generated by S + WAVELETS is shown in Fig. 4. Consistent with the notation used in this book, the cl and dl labels represent the smoothed and detailed coefficients, respectively, so that cl-cl corresponds to the smoothed (Il,l) wavelet coefficients, whereas cl-dl , dl-cl and dl-dl correspond to the detailed wavelet coefficients (Il.h, Ih.l and Ih.h sub-bands, respectively). NB In some wavelet software packages (e.g. S -l- WAVELETS ), the 

As mentioned before, when dealing with values of m greater than 2, the decomposition scheme is slightly more complicated by the fact that we have more than three sub-bands of wavelet coefficients at each level of the decomposition scheme. In fact, there are m sub-bands at each decomposition level of the 2D DWT. For the case of h levels, we have a total of (m -l)h + 1 DWT sub-bands or, for the dyadic case, a total of 3h + 1 sub- 

Whilst the 2D-DWT provides an efficient space-frequency characterisation of a given image, it only uses a fixed decomposition of the pixel space. As in the case of the 1-D wavelet packet transform, we can extend the wavelet packets to two dimensions. That is, the 2D wavelet packet transform (2D-WPT) generates a more general, full m -ary tree representation with a total of m + m + m sub-bands for h levels. Each sub-band in a given level of the tree splits into a smoothed sub-band and m -l detailed sub-bands, resulting in a tree that resembles an m-way pyramidal stack of sub-bands. For the case of a dyadic decomposition scheme, this corresponds to a pyramidal sub-band structure where each sub-band is decomposed into 2 = 4 sub-bands at each successive (higher) level (see Fig. 2). Fig. 7 shows results of the third level of the 2D WPT for the dyadic case - a total of = 64 

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Image Compression Through Wavelet Transform Coding. 

Johnston, 1996] Johnston, J. D. (1996). Audio coding with filter banks. In Akansu, A. N. and Smith, M. J. T., editors, Subband and Wavelet Transforms, pages 287 -307. Kluwer Academic Publishers. 

M. Antonini, M. Barlatid. P. Mathieu and 1. Daubechies. Image Coding using Wavelet Transform, IEEE Transactions on Image Processing. 1 (1992). 205-218. 

Abstract. We discuss the use of a wavelet transform for the subtraction of nonuniform background and for filtering images obtained with the coded-mcisk X-ray telescope ART-P aboard the Granat spacecraft. 

Key words X-ray images - coded-mask telescopes - wavelet transform... 

Describing below very briefly the results of the wavelet transform application to the Granat ART-P data, we are willing to demonstrate that this method can be also beneficially used for the analysis of X-ray images obtained with coded-mask telescopes. 

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