Combined Hierarchical Wavelet-Coefficient Structures For Grayscale Image Compression

A suitable algorithm suggested with wavelet compression for gray scale images based on one-and two-dimension combined hierarchical structure, in the sub-band wh ich has been generated by the aid of several types of wavelet functions. It is show n that the using of co mbined based hierarchical structures allows us to redu ce the calculations complexity of com pression and decompression at constant values of compression coefficients.


Introduction
Related to the intensive growth of the video traffic in networks of telecommunications, the major scientific and technical direction of development of techniques of transfer and distribution of the information are compression of video-data.The most effective algorithms are JPEG 2000, EZW, SPIHT,SPECK, which realize progressive compression of images in the field of wavelet-transformation.For achievement of high factors of compression in these algorithms, as a rule they are used the waveletfactors calculated on the basis of biorthogonal wavelet-functions.It leads to the growth of computing complexity in comparison with a variant of use for compression of wavelet-functions Haar.The purpose of the present work is the estimation of efficiency of compression of halftone pictures on the basis of the combined hierarchical structures in which wavelet-factors of various levels are generated by means of different wavelet-functions.

Rational and integer-valued wavelet-transformations
The most effective approaches for lossy image compression and lossless based on the discrete wavelet transform coding, and tree structure of wavelet coefficients, formed in the frequency-spatial domain [1 -4].Discrete wavelet transform is implemented due to the using of two types of functional approximation (approximation and detailing), applied to discrete signals at different levels of decomposition.Approximation (scaling) functions have a smooth shape envelope.Detailing of the functions (wavelet function) characterize the local features of a different order in a discrete signal (gaps, jumps, etc.).One of the most important properties of wavelet functions is the spatial and frequency localization, quantitative measure that defines the spatial-frequency resolution.The lowest spatial-frequency resolution of wavelet functions are Haar (Fig. 1, a, b), and the largest -of biorthogonal wavelets.This is due to the use of biorthogonal wavelet functions 9.7 (Fig. 1, c, d) and 5.3 (Fig. 1, e, f) in the algorithm for JPEG 2000 image compression with losses and without losses.However, the wavelet transform based on biorthogonal wavelet functions requires significantly greater computational resources than the wavelet-transform based on Haar wavelet functions.Fig. 2 presents the basic scheme of one-dimensional sound and the direct integral wavelet transform based on Haar wavelet functions and biorthogonal wavelet functions, which are formed as a result of a low (L) and one high (H) wavelet coefficients.
Fig. 2, a and 2, b shows that the basic rational wavelet transform based on biorthogonal wavelet functions requires 9.7 to 4 times more multiplicative operations and 7 times more than double additive operations compared to the rational wavelet transform based on Haar wavelet function.From Fig. 2, c and 2, d it is shown that the basic integer-valued wavelet transform based on biorthogonal wavelet functions requires 5.3 to 2 times more than multiplicative and additive operations, compared with an integer-valued wavelet transform on Haar wavelet functions.

Hierarchical structure of the wavelet coefficients
In the frequency domain the wavelet transform can be represented as a bank of filters.Fig. 3 shows three-cascaded filter bank analysis and synthesis, carrying out a direct and inverse wavelet transform sound.multilevel discrete wavelet decomposition of signals obtained by recursive application of the lowfrequency and high frequency filtering to the original values of the signal at the first iteration and the low wavelet coefficients at subsequent iterations.The direct and inverse wavelet transforms sound described by the following recurrent expression: PDF created with pdfFactory Pro trial version www.pdffactory.com where F φ and F ψ -direct rational wavelet transform using the approximation, and detailing the wavelet functions, 1 the number of wavelet decomposition, coinciding with the number of iterations ,R the number of levels of wavelet decomposition, ( )  The tree structure is obtained by combining the wavelet coefficients at all levels of transformation and accession to the low-frequency coefficients of the last level (Fig. 4).
One dimensional tree structure includes a degenerate tree of L (approximation ratio) and the tree H (detailing the factors).When building two tree structures using a one-dimensional wavelet transform applied to the first row and then two columns to the matrix of input data or the wavelet coefficients.In doing so, formed four sub-band coefficients (LL -approximation ratio; HL, HH, LH -detailing coefficients).Similarly, we construct three-dimensional tree structure, which includes eight sub-bands.The number of wavelet coefficients in the tree structure is equal to the number of discrete and dimension (the dimension of the space) tree PDF created with pdfFactory Pro trial version www.pdffactory.com( ) combined treelike structure, using Haar wavelet functions on the lower level, biorthogonal wavelet functions 9.7 and 5.3 m at the following levels and Haar wavelet functions on the upper level n .The F and I are subscripts, in the notation of tree structures to indicate the type of wavelet transform in rational or integervalued.From the table. 1 shows that the combined use of trees instead of a uniform tree-based biorthogonal wavelet functions reduces the multiplicative (additive) computational complexity in the 38-42% (43-48%) for the management of wavelet transform and 23-28% (25 -28%) for the integer-valued wavelet transform.

Valuation descriptions compactness combined trees
The effectiveness of combined trees can be evaluated by using algorithms of wavelet compression, lossy and lossless.In the tables 2 and 3 they are shown the characteristics of compression, lossy and lossless for standard grayscale test images «Lena» (low) and «Barbara» (a high) pixels in size, obtained through a one-dimensional version of the algorithm MECT [4]  B .When compression of 16 and 32 times, using a combined tree-structure on the reconstruction of the image block effect and no small parts saved about the same as when using a homogeneous structure . The using of the tree structure provides a reduction of computational complexity of approximately 1.7fold compared with the tree (a branch of hierarchal) structure 9 F B .Fig. 5 and 6 show the reconstructed test image «Lena» and «Barbara», compressed to 0.0625 bits per pixel (picture «Lena») and up to 0,125 bits per pixel (picture «Barbara») using two-dimensional algorithm MECT wavelet-based structures, indicated in Table .4. From Fig. 5 and  ).

Conclusions
The proposed algorithm has been used for formation of onedimensional multi-layered combined trees of wavelet coefficients at the lower levels which the wavelet transform based on spatial-frequency localized biorthogonal wavelet functions, and on the upper level, a simple computational wavelet transform based on Haar wavelet functions.We show that in comparison with similar trees based on biorthogonal wavelet functions, the combined use of trees reduces the computational complexity of one-dimensional compression of grayscale images by approximately 40% for the management of wavelet transform and roughly 30% for integer-valued wavelet transformation while maintaining or improving peak signal-noise ratio and compression ratio.Using twodimensional hierarchal structure of wavelet coefficients provides a reduction of computational complexity up to 1.7 times compared with homogeneous biorthogonal 9.7 wavelet structure with preservation of the quality of image reconstruction.

References
of the low-and high-frequency filters at the level r of decomposition () on small parts of images, but on top -Haar wavelet function for high spatial resolution and a compact representation of the major details of the prevailing in the low-frequency area of the upper levels of wavelet decomposition.
and JPEG 2000 , Proc. of the SPIE, San Diego, PDF created with pdfFactory Pro trial version www.pdffactory.com