Two-dimensional integer wavelet transform with reduced influence of rounding operations
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RESEARCH
Open Access
Two-dimensional integer wavelet transform with reduced influence of rounding operations Tilo Strutz* and Ines Rennert
Abstract If a system for lossless compression of images applies a decorrelation step, this step must map integer input values to integer output values. This can be achieved, for example, using the integer wavelet transform (IWT). The non-linearity, introduced by the obligatory rounding steps, is the main drawback of the IWT, since it deteriorates the desired filter characteristic. This paper discusses different methods for reducing the influence of rounding in 5/3 and 9/7 filter banks. A novel combination of two-dimensional implementations of the JPEG2000 9/7 filter bank with new filter coefficients is proposed and the effects of the methods on lossless image compression are investigated. In addition, these filter banks are compared to the 9/7 Deslauriers-Dubuc filter bank (97DD). The analysed two-dimensional implementations generally perform better than their one-dimensional counterparts in terms of compression ratio for natural images. On average, the 2D 97DD filter bank performs best. In addition, it has been found that the compression results cannot be improved by simply reducing the number of lifting steps via 2D implementations of the JPEG2000 9/7 filter bank. Only the 2D implementation with a minimum number of lifting steps, in combination with modified lifting coefficients, leads to fewer bits per pixel than the separable implementation on average for a selected set of images. Keywords: integer wavelet transform; filter bank; 2D lifting; image compression; rounding
1 Introduction Efficient systems for the lossless compression of image data require a decorrelation step which maps the integer input samples to integer output values. In wavelet-based compression systems (see [] for an overview), this is achieved by using the lifting implementation of a discrete wavelet transform (DWT) [] in combination with the rounding of intermediate computation results, which is called integer wavelet transform (IWT) []. Beginning with initial investigations on the IWT, which were also motivated by the standardization of the new image compression system JPEG [], and its application in the JPEG framework [], this topic has received growing attention. The idea of integer transforms relates back to the so-called S * Correspondence: [email protected] Deutsche Telekom, Hochschule für Telekommunikation Leipzig, Institute of Communications Engineering, Gustav-Freytag-Str. 43-45, Leipzig, 04277, Germany
transform [], the improved version called S + P transform [], and the reversible TS transform []. Since then, several integer wavelet transforms have been analysed in terms of their performance in image compression systems []. Wavelet filter banks are typically designed without taking the integer-to-integer mapping into account. The conversion into an integer wavelet transform requires rounding steps which introduce non-linear effects, which deteriorate the desired filter properties. That is on
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