Fixed Wordsize Implementation of Lifting Schemes
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Research Article Fixed Wordsize Implementation of Lifting Schemes Tanja Karp Department of Electrical and Computer Engineering, College of Engineering, Texas Tech University, P.O. Box 43102, Lubbock, TX 79409-3102, USA Received 16 December 2005; Revised 29 May 2006; Accepted 26 August 2006 Recommended by Soontorn Oraintara We present a reversible nonlinear discrete wavelet transform with predefined fixed wordsize based on lifting schemes. Restricting the dynamic range of the wavelet domain coefficients due to a fixed wordsize may result in overflow. We show how this overflow has to be handled in order to maintain reversibility of the transform. We also perform an analysis on how large a wordsize of the wavelet coefficients is needed to perform optimal lossless and lossy compressions of images. The scheme is advantageous to well-known integer-to-integer transforms since the wordsize of adders and multipliers can be predefined and does not increase steadily. This also results in significant gains in hardware implementations. Copyright © 2007 Tanja Karp. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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INTRODUCTION
Lifting schemes have been introduced by Daubechies and Sweldens as a structure to design and implement the discrete wavelet transform (DWT) [1]. They have gained immense popularity, since they provide an elegant means to maintain perfect reconstruction of the DWT while staying in the field of integer numbers [2], thus allowing for lossless coding. At each lifting step, a quantizer that ensures that intermediate results are rounded to integer numbers is introduced. Lifting schemes have been applied to a variety of fields. They are used to implement the polyphase filters of modulated filter banks [3], are building blocks of integer-to-integer transforms such as the IntFFT [4], the IntDCT [5], and YCoCg-R color space transform [6]. More recently, the concept of lifting has been extended to multidimensional inputs [7]. Lifting schemes can even be designed in such a way that they preserve signal symmetries, a fact that is important for image compression. The performance of nonlinear reversible DWT using lifting schemes has been studied in [8]. While keeping the wordlength of the wavelet coefficients finite is a necessary condition for lossless coding, the dynamic of the signal and thus its wordlength normally increase. This is not only disadvantageous from a compression point of view, but also means a higher complexity of the circuitry. For example, in an FPGA, the number of gates needed for a multiplier increases significantly with the wordlength. It is therefore of interest to keep the maximum wordsize limited, thus resulting in a fixed-point implementation of the
lifting scheme, where, in addition to rounding, overflow occurs if a value exceeds the range of representable numbers. In this paper, we show that lifting schemes are also robust toward
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